UNIVEKSITY  OF  CALIFOIu.IA 

LI BEAKY 
D3PAR2.IEIIT  0?  CIVTL 


Gift   of  Lirs.   Edwin  H.  Warner  from 
her  husband fs   library. 


January     1928 


Library 


UNIVERSITY  ©F  CAUia'01*NIA 

OF  CIVIL  ENGINEERING 


UNIVERSITY  or 

DEPARTMENT  OF  CIVIL 

,  CAL.IFOr?MIA 


ELEMENTS    OF 
ELECTROMAGNETIC    THEORY 


• 


ELEMENTS 


OF 


ELECTROMAGNETIC  THEORY 


BY 


S.  J.  BARNETT,  PH.D. 
;/ 

ASSISTANT  PROFESSOR  OF  PHYSICS 

IN   THE 

LELAND  STANFORD,  JR.,  UNIVERSITY 


gork 

THE    MACMILLAN    COMPANY 

LONDON  :    MACMILLAN  &  CO.,  LTD. 

1903 

All  rights  reserved 


tins  Peering 
Library 

Copyright,  1903 
BY  THE  MACMILLAN    COMPANY 

Set  up,  electrotyped  and  printed  September,  1903 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANV. 
LANCASTER.  PA. 


CORRIGENDA. 

Page  4,  line  i6,for  unelectrified  (sixth  word}  read  electrified 

Page  33»  lme  io,/<?r  intensity  read  tension 

Page  127,  line  i,for  Ad2  read  d2 •  A 

Page  128,  line  7,  insert  5.  at  beginning  of  line. 

Page  130,  line  8,  for  5  read  6      .. 

Page  203,  lines  5,  6-7,  24,  cancel  isotropic 

Page  204,  line  \,for  ^  -  V2  read  Vu 

Page  212,  lines  6-7,  cancel  the  expression  in  brackets. 

Page  216,  lines  27  and  30,  cancel  (39)  and  (40) 

Page  221,  line  28,  for  V^  -  V2  =  ¥12  rao/  F12  =  ¥21 

Page  244,  line  4,  ^r  to  2;^;^  (one  form  of) 

Page  290,  line  14,  for  3  read  6 

Page  315,  line  22,  for  25  read  26 

Page  426,  line  9,  ^/fcr  and  /;/^r/  even  for  small  values  of  r 

Page  426,  line  10,  for  comparable  with  read  greater  than  a 
small  fraction  of 

Page  441,  line  29,  for  (9)  read  (a) 

Page  450,  lines  18-19,  substitute  small  bodies  with  equal  and 
opposite  charges  are  made  to  vibrate  symmetrically  with  (ap- 
proximately) simple  harmonic  motion  in  a  straight  line  about  a 
fixed  point,  a  wave  system 


793211 


DEDICATED 

WITH 

GRATITUDE  AND  AFFECTION 

TO 

MY   REVERED   FRIEND 

PROFESSOR   FRANCIS    H.    SMITH,    LL.D. 


UNIVERSITY  OF  VIRGINIA 


PREFACE. 

In  this  treatise  I  have  tried  to  present  in  systematic  and  defi- 
nite form  a  simple,  rigorous,  and  thoroughly  modern  introduc- 
tion to  the  fundamental  principles  of  electromagnetic  theory, 
together  with  some  of  the  simpler  of  their  more  interesting  and 
important  non -technical  applications.  The  work  makes  no  pre- 
tense to  completeness,  but  is  written  for  the  serious  student  of 
physics,  who  will  make  liberal  use  of  more  detailed  treatises,  of 
hand-books,  and  of  journals,  as  occasion  demands. 

I  am  of  course  indebted  to  many  books  and  memoirs.  My 
obligations  are  especially  great,  as  the  most  cursory  examination 
of  the  book  will  show,  to  the  works  of  Maxwell,  Heaviside,  and 
Poynting.  I  am  also  much  indebted  to  Professor  A.  G.  Webster 
for  the  use  of  a  number  of  excellent  diagrams  from  his  treatise 
on  electrical  theory. 

S.  J.  BARNETT. 

STANFORD  UNIVERSITY,  CALIFORNIA 
June,  1903. 


CONTENTS. 

CHAPTER.  PAGE. 

I.     GENERAL  ELECTROSTATIC   THEORY i 

II.     IDEAL   ELECTRIC    FIELDS    AND    CONDENSERS    WITH    HOMO- 
GENEOUS  Di  ELECTRICS 57 

III.  STANDARD  CONDENSERS.     CONDENSER  SYSTEMS.     ELECTROM- 

ETERS      122 

IV.  ELECTRIC  FIELDS  WITH  Two  OR  MORE  DIELECTRICS 139 

V.     REVERSIBLE   THERMAL    EFFECT.     ELECTROSTRICTION 168 

VI.     ELECTRIC  ABSORPTION.     ELECTRETS 176 

VII.     COMPARISON  OF  DIELECTRIC  CONSTANTS.     SPECIFIC    INDUC- 
TIVE CAPACITY 192 

VIII.     THE    ELECTRIC   CONDUCTION   CURRENT.      INTRINSIC   ELEC- 
TROMOTIVE  FORCE 199 

IX.     ELECTROLYTIC  AND  METALLIC  CONDUCTION 228 

X.     THERMAL  AND  VOLTAIC    ELECTROMOTIVE   FORCES 246 

XL     MAGNETS.     MAGNETOSTATIC    FIELDS 265 

XII.     THE  MAGNETIC  FIELD  OF  THE  CONDUCTION  CURRENT 286 

XIII.  ELECTROMAGNETIC  INDUCTION 332 

XIV.  UNITS  AND   DIMENSIONS 415 

XV.     CONVECTION  AND  DISPLACEMENT  CURRENTS.     THE  GENERAL 

ELECTRIC  CURRENT 424 

XVI.     THE     FLUX     OF    ELECTROMAGNETIC    ENERGY.      ELECTRIC 

WAVES 433 


• 
•r 


ELEMENTS  OF 
ELECTROMAGNETIC  THEORY. 


CHAPTER    I. 
GENERAL   ELECTROSTATIC    THEORY. 

V  electrification  by  Contact.  Positive  and  Negative  Charges. 
Let  one  end  of  an  ebonite  rod  and  a  dry  woolen  cloth  be  rubbed 
01  rongly  pressed  together  and  then  separated  ;  and  let  a  second 
roc  nd  cloth  be  treated  in  the  same  way  :  The  rubbed  part  of 
ea  .  cloth  will  be  found,  on  trial,  to  be  attracted  toward  the 
n  d  part  of  each  rod,  while  the  rubbed  part  of  each  cloth  will 
be  pelled  from  the  rubbed  part  of  the  other  cloth,  and  the 
i  jd  part  of  each  rod  from  the  rubbed  part  of  the  other  rod. 

.  .ese  are  examples  of  electric  phenomena.  The  region  in 
which  they  are  manifested  is  called  an  electric  field  (§  1 1),  and 

medium  which  permeates  this  region  —  air  and  aether  in  the 
al  e  case  —  and  through  which  electric  influences  are  trans- 
r  ''  xl  is  called  a  dielectric.  The  parts  of  the  ebonite  and  wool 

jed  together  are  said  to  be  electrified,  or  to  possess  electric 
'rges.  The  two  pieces  of  woolen  cloth  are  said  to  have  like 

rges,  since  they  were  similarly  treated  and  since  what  is 
.pelled  from  one  is  repelled  from  the  other,  and  what  is  attracted 
toward  one  is  attracted  toward  the  other.  Similarly,  the  two 
ebonite  rods  are  said  to  have  like  charges.  But  the  wool  and  the 
ebonite  are  said  to  have  unlike  or  opposite  charges,  since  what  is 
repelled  from  one  is  attracted  toward  the  other. 


I-L^AtEATS    OF    ELECTROMAGNETIC    THEORY. 

Like  ebonite  and  wool,  any  two  different  substances,  or  por- 
tions of  the  same  substance  in  different  physical  conditions, 
exhibit  electric  properties  after  intimate  contact  and  separation. 
One  of  the  bodies  behaves  like  ebonite  rubbed  with  wool,  the 
other  like  the  wool. 

An  electric  charge  like  that  of  wool  after  contact  with  ebonite 
is  called  a  positive  charge,  and  a  charge  like  that  of  the  ebonite, 
a  negative  charge.  The  terms  positive  and  negative  are  justified 
by  the  opposite  properties  of  the  two  kinds  of  electrification,  but 
there  is  no  reason  except  convention  and  resulting  convenience 
why  the  two  terms  should  not  be  interchanged. 

In  addition  to  the  forces  between  electrified  bodies,  forces  are 
found  to  exist,  in  general,  between  an  electrified  body  and  an 
insulator  (§  2)  not  electrified  (Chapters  IV.  and  VI.). 

2.  Conductors  and  Insulators.  Electrification  by  Conduction. 
A  rod  of  ebonite  electrified  at  one  end  exhibits  electric  properties 
only  at  that  end ;  while  a  rod  of  metal,  held  by  an  ebonite 
handle  and  electrified  at  one  end,  becomes  electrified  at  once 
(apparently)  all  over  its  surface.  Substances  like  the  metals,  by 
which  an  electric  charge  is  distributed  with  extreme  rapidity,  so 
as  to  come  into  a  state  of  equilibrium  within  (usually)  a  small 
fraction  of  a  second,  are  called  electric  conductors.  A  body 
charged  by  connection  with  an  electrified  body  through  a  con- 
ductor, like  the  far  end  of  the  metal  rod  mentioned  above,  is  said 
to  be  electrified  by  conduction.  Substances  like  ebonite,  over  or 
through  which  an  electric  charge  is  transferred  only  with  extreme 
slowness,  are  called  electric  insulators  or  non-conductors. 

Among  ordinary  molecular  substances  perfect  insulators  and 
perfect  conductors  do  not  exist,  no  such  substance  completely 
and  for  an  indefinite  time  preventing  all  transfer  of  electrification, 
and  all  offering  more  or  less  obstruction  to  such  transfer.  There 
is  every  reason  to  believe,  however,  that  free  aether  (a  "  vacuum  ") 
and  clean  dry  gases  containing  no  (electrolytically)  dissociated 
molecules  have  the  properties  of  a  perfect  insulator  (Chapter  IX.). 


GENERAL  ELECTROSTATIC  THEORY.          3 

Among  substances  possessing  high  conductivity  are  the  metals, 
graphite,  and  salt  or  acid  solutions  ;  among  those  with  high  in- 
sulating properties  are  (undissociated)  gases,  fused  quartz  (cold 
and  in  the  solid  state),  ebonite,  cold  glass,  silk,  and  wool.  A 
substance  which  is  an  excellent  insulator  in  one  condition,  how- 
ever, may  in  another  condition  have  the  properties  of  a  conductor. 
Thus  cold  glass  is  an  excellent  insulator,  but  as  the  temperature 
is  raised  its  insulating  properties  disappear.  Also,  under  very 
great  electric  stress  the  insulating  properties  of  all  molecular 
substances  break  down. 

A  body  completely  surrounded  with  insulators  is  said  to  be 
insulated. 

A  conductor  can  be  completely  discharged  by  bringing  it  into 
contact  at  any  one  point  with  the  inner  surface  of  a  hollow  closed 
conductor  (§  4),  such  as  the  walls  of  the  room  within  which  the 
experiments  are  performed,  provided  there  are  no  (insulated) 
electrified  bodies  within.  When  connected  to  the  walls  of  the 
room,  or  the  earth,  the  conductor  is  said  to  be  earthed.  From 
an  insulator  the  electrification  can  be  entirely  removed  only  by 
applying  a  conductor  at  every  electrified  point,  e.  g.,  by  immers- 
ing it  in  a  conducting  gas  or  liquid. 

3.  Electrification  by  Induction.  An  insulated  conductor,  when 
brought  near  an  electrified  body,  i.  e.,  into  an  electric  field,  itself 
becomes  electrified.  Examined  by  the  methods  of  §  I,  the 
charges  of  the  more  remote  and  nearer  ends  of  the  conductor  are 
found  to  be  similar  and  opposite,  respectively,  to  that  of  the 
original  electrified  body.  A  conductor  electrified  in  this  manner 
is  said  to  be  electrified  by  induction. 

If  the  conductor,  while  still  insulated,  is  removed  from  the 
electric  field,  all  signs  of  electrification  disappear.  But  if,  while 
still  in  the  field,  it  is  connected  with  the  walls,  or  earthed,  the 
electrification  similar  to  that  of  the  original  charged  body  disap- 
pears, while  the  opposite  electrification  of  the  near  end  remains. 
If  the  conductor  is  now  insulated  and  removed  from  the  original 


4  ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

electric  field,  this  charge  becomes  more  evenly  distributed  over 
its  surface  (§  42).  In  this  manner  any  number  of  conductors 
may  be  given  charges  opposite  to  that  of  a  given  electrified  body 
without,  as  may  be  proved  by  the  method  of  §  5,  diminishing  or 
increasing  the  latter' s  electrification. 

4-8.  Experiments  with  Hollow  Closed  Conductors.  Electric 
Screens.  Let  A  denote  an  insulated  hollow  conductor  having  a 
closely  fitting  conducting  lid,  B,  with  an  insulating  handle.  Let 
A  be  connected  with  an  electroscope  or  electrometer  (Chapter 
III.),  C,  by  means  of  which  any  change  in  the  state  of  electri- 
fication of  its  exterior  (or  interior)  surface  may  be  detected  ;  and 
let  A  be  kept  closed  except  when  another  body  is  being  intro- 
duced into  its  cavity,  or  removed  therefrom,  or  its  position  in- 
side (or  outside)  altered. 

4.  (i)  Let  the  electrometer  be  placed  outside  of  A.  If  A  is 
initially  unelectrified,  and  an  insulated  ^electrified  conductor,  D, 
is  now  introduced  into  A  without  touching  it,  the  inner  and  outer 
surfaces  of  A  will  become  electrified  by  induction  (§3)  with 
charges  opposite  and  similar,  respectively,  to  that  of  D.  And 
the  electrification  of  the  external  surface,  as  indicated  by  the 
electrometer,  will  be  found  to  remain  absolutely  unaltered  how- 
soever D  is  moved  about  within,  even  when  it  is  brought  into 
contact  with  A  ;  but  D,  on  being  insulated  after  contact,  and  then 
removed  from  A's  interior,  will  be  found  completely  discharged. 
This  process  may  be  repeated  indefinitely,  D  always  becoming 
completely  discharged  on  coming  into  contact  with  the  inner 
surface  of  A.  If  A  is  initially  electrified  in  any  manner,  the  phe- 
nomena will  be  precisely  the  same,  except  that  the  external 
electrification  and  the  corresponding  indication  of  the  electrom- 
eter will  be  different. 

(2)  Let  the  electrometer  be  placed  within  A,  either  connected 
with  A  metallically,  or  insulated  therefrom.  In  this  case  it  will 
be  found  that  if  there  are  insulated  charged  bodies  within  A,  the 
electrometer  will  give  a  certain  deflection  ;  that  if  there  are  no 


GENERAL  ELECTROSTATIC  THEORY.          5 

insulated  electrified  bodies  within  A,  the  electrometer  will  give 
no  deflection ;  and  that  its  indication  in  either  case  will  remain 
absolutely  unaltered  howsoever  the  electrification  of  the  exterior 
of  A  or  of  external  bodies  is  changed,  even  if  A  is  connected  to 
the  walls  of  the  room. 

These  experiments  are  due  to  Faraday,  who  constructed  for  the 
purpose  of  performing  (2)  a  closed  conductor  large  enough  to  en- 
able him  to  make  the  observations  while  himself  inside  the  cavity. 

An  experiment  similar  in  principle  to  those  of  Faraday,  but 
less  general,  performed  earlier  by  Cavendish  and  repeated  later 
by  Maxwell  with  all  the  precision  of  modern  investigation,  gave 
identical  results. 

From  the  experiments  just  described  it  follows  that,  when 
there  is  electrical  equilibrium, 

1.  An   electric  charge  cannot  exist  in  the  substance  of  a  con- 
ductor,   or  on  the   inner  surface    of    a   hollow   closed  conductor 
(unless  there  are  insulated  electrified  bodies  within).      For  D,  on 
being  removed  from  A,  of  whose  substance  it  formed  a  part, 
electrically,  while  in  contact,  was  always  unelectrified. 

2.  An   electric  field  (§   11)  does  not  exist  within  the   hollow 
of  a  closed  conductor  (unless  there  are  charges  inside).      For  in 
(2)  the  electrometer  was  unaffected  (by  induction  or  otherwise) 
no  matter  what  the  external   electrification,  except  when  there 
were  insulated  charges  within. 

3.  The  electric  charges  and  electric-  fields  within  and  without  a 
hollow  closed  conductor  are  absolutely  independent  of  one  another. 
The  conducting  shell  thus  completely  screens  each  of  these  re- 
gions from  all  static  effects  in  the  other. 

4.  An  electric  field  does  not  exist  within  the  substance  of  a  con- 
ductor.    See  §  15. 

5.  Equal  Charges.     Two   electric  charges  of   the   same  sign 
are,  by  definition,  of  the  same   magnitude  if  they  produce  the 
same  effect  on   the  electrification  of  the  vessel  A  when  intro- 
duced in  succession  separately. 


6  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

Similarly,  two  charges  of  opposite  signs  are,  by  definition, 
equal  in  magnitude  if  they  produce  no  effect  on  the  electrification 
of  A  when  introduced  simultaneously. 

These  definitions  are  independent  of  the  particular  closed 
conductor  A  used,  as  two  charges  defined  as  equal  by  means  of 
one  such  vessel  are  found  to  remain  equal  when  tested  in  the 
same  way  with  any  other  hollow  closed  conductor. 

6.  Positive    and   Negative    Charges    are    Always    Developed 
Simultaneously  in  Equal  Amounts.      If  two  bodies  electrified  by 
contact  are  introduced  into  the  vessel  A  simultaneously,  the  in- 
dication of  the  electrometer  remains  unaltered. 

If  an  electrified  body  is  insulated  within  At  and  if  an  insulated 
uncharged  conductor  is  then  introduced  in  addition,  the  latter 
becomes  electrified  by  induction,  in  conformity  with  §  3,  but  the 
indication  of  the  electrometer  remains  unaltered. 

In  these  cases,  therefore,  positive  and  negative  charges  are 
developed  in  equal  amounts  (§  5);  and  in  the  same  way  it  may 
be  shown  that  this  is  always  the  case,  howsoever  the  electrifica- 
tion is  produced. 

7.  The  Total  Quantity  of  Electrification  is  Unaltered  by  Con- 
duction.     If  the  two  insulated  bodies  of  the  last  experiment  are 
brought  into  contact  with  one  another  while  inside  the  vessel  A,  or 
if  they  are  brought  into  contact  with  the  inner  surface  of  A  itself, 
conduction  occurs,  but  no  effect  on  the  external  electrification  is 
produced.      From  this  it  follows  that  when   conduction  occurs, 
the  total  (algebraic)  amount  of  electrification  is  unaltered. 

Corollary.  The  charges  induced  on  the  inner  and  outer  sur- 
faces of  A  when  an  electrified  body  is  introduced  and  insulated 
within,  as  in  §  4,  are  each  of  the  same  magnitude  as  that  of  the 
visulated  body.  For  when  D  touches  A,  the  charges  of  D  and 
the  inner  surface  of  A  completely  disappear  by  conduction,  since 
D  on  removal  is  unelectrified  ;  thus  their  algebraic  sum  is  zero. 
And  the  (opposite)  charges  on  the  inner  and  outer  surfaces, 
being  induced,  must,  by  §  6,  be  equal  in  magnitude. 


GENERAL  ELECTROSTATIC  THEORY.          7 

8.  Electric  Charges  of  Both  Kinds  Measured  in  Terms  of  a 
Single  Arbitrary  Unit.  In  addition  to  the  hollow  conductor  A 
of  §  §  4-7,  let  there  be  provided  another  similar  insulated  vessel 
B,  sufficiently  large  to  admit  A  through  its  opening ;  and  let  the 
conductor  D  be  given  a  certain  charge  (suppose  positive  for  the 
sake  of  definiteness),  which  will  be  adopted  as  a  provisional  unit. 

If  now  D  is  brought  within  A  and  kept  insulated,  the  outer 
surface  of  A  will  have  unit  positive  charge.  If  A  is  brought  in- 
side B  and  then  into  contact  with  it,  this  charge  will  disappear,  as 
will  also  the  charge  induced  on  B's  inner  surface,  leaving  the 
outside  of  B  with  unit  positive  charge.  If  A  is  now  removed 
from  j5's  interior  and  then  D  from  A,  the  negative  charge  in- 
duced on  A's  inner  surface  will  pass  to  the  outer  surface  and 
will  disappear  when  A  is  discharged.  This  complete  process 
may  be  repeated  any  number  of  times.  Each  time  B  will  acquire 
an  additional  unit  positive  charge,  and  thus  may  be  given  a 
measured  positive  charge  which  is  any  integral  multiple  of  the 
original  unit. 

To  give  B  a  negative  charge  measured  in  terms  of  the  same  unit, 
the  outer  surface  of  A  must  be  brought  into  contact  with  the 
inner  surface  of  a  hollow  closed  conductor  after  the  introduction 
of  D,  when  the  positive  charge  will  disappear  from  the  outside, 
leaving  unit  negative  charge  upon  the  inner  surface.  When  D 
is  removed,  this  charge  will  pass  to  the  outer  surface  of  A,  and 
will  be  given  up  wholly  to  B  when  A  is  brought  into  contact 
with  B's  interior.  B  will  now  have  unit  negative  charge,  and  by 
removing  A  and  repeating  the  process  may  be  given  any  number 
of  units  negative  charge  desired. 

To  obtain  any  submultiple,  i/«,  of  the  original  charge,  it  is 
only  necessary  to  arrange  symmetrically  in  contact  the  original 
conductor  D  and  n  —  I  precisely  similar  and  equal  conductors, 
all  other  bodies,  except  the  surrounding  dielectric,  supposed 
homogeneous  and  isotropic,  being  so  remote  as  to  have  no  appre- 
ciable effect.  Then,  by  the  principle  of  symmetry,  each  con- 
ductor will  take  ijn  of  the  original  charge. 


8  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

9.  The  Law  of  Coulomb.  Let  two  small  spherical  insulated 
conductors  which  can  be  given  any  charge  desired,  measured  in 
terms  of  some  provisional  unit  by  the  methods  of  §§5  and  8, 
be  so  connected  with  a  dynamometer,  such  as  a  gravity  balance, 
that  the  force  F  between  them  can  be  measured  as  their  charges, 
ql  and  qv  the  distance  L  between  their  centers,  and  the  surround- 
ing dielectric  are  varied.  ,  Then  it  is  found  by  experiment  that? 

(1)  However  the  distance  L  and  the  charges  gl  and  qz  are 
varied,  provided  all  the  experiments  are  performed  in  the  same 
dielectric,  and  provided  that  this  dielectric  is  homogeneous  and 
isotropic  and  extends  to  a  great  distance  on  all  sides  of  the  elec- 
trified bodies,  F  is  in  the  straight  line  joining  the  centers  of  the 
conductors ;    is   directly   proportional   to    the    product   of  their 
charges,  being  repulsive  (considered  positive)  when  the  charges 
are  like  and  attractive  (considered  negative)  when  the  charges 
are  unlike,  as  already  known  from  §  I  ;  and  the  greater  L  in  com- 
parison with  the  linear  dimensions  of  the  charged  bodies,  the 
more  nearly  inversely  proportional  to  Z2. 

(2)  In  different  dielectrics,  with  all  other  conditions  the  same, 
the  force  is  different,  and  always  less  than  in  vacuo  (free  aether). 

The  general  expression  for  Ft  when  the  linear  dimensions  of 
the  (not  necessarily  spherical)  charged  bodies  are  negligible  in 
comparison  with  their  distance  apart,  is  therefore 

•-  ::'          F-AqjJcD    :       '  (i') 

where  c  is  a  constant  depending  on  the  medium  in  which  the  ex- 
periments are  performed,  called  its  permittivity  or  dielectric  con- 
stant, and  A  is  a  positive  constant  depending  on  the  units  in 
which  qv  q2,  Ly  F,  and  c  are  expressed, 
(i')  expresses  Coulomb 's  law -. 

The  Rational  Electrostatic  Unit  Charge.  Unit  Permittivity.  In 
what  follows,  unless  the  contrary  is  stated,  the  centimeter  will 
be  used  as  unit  length,  the  dyne  as  unit  force,  the  permittivity 
of  free  aether,  which  will  be  denoted  by  CQ,  as  unit  permittivity, 


GENERAL  ELECTROSTATIC  THEORY.          9 

and  as  unit  charge  the  charge  which  each  of  two  indefinitely 
small  bodies  must  have  in  order  that  when  at  a  distance  of  I  cm. 
apart  in  a  vacuum  the  force  between  them  may  be  1/477  dyne. 
This  unit  charge  is  called  by  its  originator,  Oliver  Heaviside,  the 
rational  electrostatic  unit  charge,  and  CQ  is  called  the  electrostatic 
unit  permittivity. 

Methods  of  measuring  permittivity  are  discussed  in  Chapter 
VII. 

The  conventions  just  made  give,  by  the  above  equation, 
A  =  I/47T,  and  the  equation  reduces  to 

F '  =  qfalcqirL*  (l) 

which,  in  addition  to  being  a  particular  case  of  (V),  is  a  particular 
case  of  (2). 

The  direct  experimental  investigation  of  the  law  of  force  is 
due  to  Coulomb,  but  is  not  capable  of  great  precision.  The  law, 
as  stated  by  Coulomb,  is  most  satisfactorily  established  by  the 
consideration  that  all  experimental  knowledge  is  in  perfect  accord 
with  an  electrical  theory  based  largely  upon  the  assumption  that 
the  laws  expressed  in  (i)  are  exact.*  A  reason  for  the  law  of 
inverse  squares  and  a  justification  of  the  term  rational  unit  will 
be  given  in  §§  5,  II.,  and  24. 

The  dimensions  of  electric  charge  and  the  other  electric  quan- 
tities, as  well  as  other  systems  of  units,  will  be  considered  in 
Chapter  XIV. 

For  rational  electrostatic  the  abbreviation  RES  will  hereafter 
be  employed. 

10.  If  any  one  of  the  experiments  described  above  is  repeated 
in  different  dielectrics,  the  results  in  all  cases  will  be  identical, 
except  that,  in  conformity  with  §  9,  the  force  between  two 
charged  bodies  will  always  depend  on  the  surrounding  dielec- 
tric. 

*The  common  deduction  of  the  law  of  inverse  squares  from  the  results  of  the 
Cavendish  experiment  cannot  be  accepted  as  valid.  See  The  Physical  Review,  Sep- 
tember, 1902,  p.  175. 


IO  ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

11.  Electric  Field,  Electric  Intensity. — Any  region  in  which 
an  electrified  body  is  acted  upon  by  a  mechanical  force  in  virtue 
of  its  charge,  or  in  which  an  uncharged  conductor  is  charged  by 
induction,  is  called  an  electric  field.  Such  a  field  exists,  for  ex- 
ample, around  an  electrified  body  (§  i),  but  may  also  exist  with- 
out the  presence  of  electrification  (Chapters  VI.  and  XIII. ). 

As  a  result  of  experiment,  it  may  be  stated  that  the  force  F 
acting  upon  a  small  charged  body,  or  small  portion  of  a  charged 
body,  at  any  point  of  an  electric  field  is  proportional  to  its  charge 
q  —  provided  that  the  distribution  of  electric  charge  (real  and 
apparent,  Chapter  IV.)  originally  accompanying  the  electric  field 
remains  undisturbed  by  the  introduction  of  q.  Expressed  in  the 
form  of  an  equation,  this  relation  is 

F-  Eq  (2) 

where  £  is  a  constant  for  the  given  point  of  the  field  called  the 
electric  intensity,  electric  force,  or  voltivity  at  the  point. 

The  conditions  for  the  rigorous  proof  of  this  relation  by  direct 
experiment  would  be  impossible  to  realise,  and  the  remark  at  the 
close  of  §9  with  reference  to  the  establishment  of  Coulomb's  law 
applies  without  alteration  to  (2). 

As  (2)  shows,  E  is  not  a  mere  number,  but  a  physical  quan- 
tity specifying  the  state  of  the  field  and  such  that  its  product  by 
an  electric  charge  is  a  mechanical  force.  E  is  clearly  a  vector 
quantity,  its  direction  being  that  of  the  force  on  a  positively 
charged  body,  and  its  magnitude  the  number  of  dynes  per  unit 
charge.  When  q  is  expressed  in  the  RES  unit  charge  and  F  in 
dynes,  E  is  said  to  be  expressed  in  the  RES  unit  electric  inten- 
sity. 

The  term  electric  field  is  often  used  to  denote  the  collective  in- 
tensity in  a  region,  instead  of  the  region  itself.  The  direction  of 
the  field  at  any  point  is  the  direction  of  the  intensity,  and  the 
strength  of  the  field  \s  the  magnitude  of  the  intensity. 

12.  The  Superposition  of  Electric  Fields.  —  Experiment  also 
shows  that  any  number  of  electric  fields  (up  to  a  certain  limit, 


GENERAL  ELECTROSTATIC  THEORY.         II 

when  the  dielectric  breaks  down  and  conduction  occurs)  may  be 
superposed  upon  one  another,  the  effect  of  each  being  indepen- 
dent of  all  the  rest.  Electric  intensities,  being  vectors,  may 
therefore  be  compounded  like  all  other  vectors  for  which  the 
principle  of  superposition  holds,  the  resultant  intensity  at  any 
point  being  the  geometric  or  vector  sum  of  the  component  in- 
tensities. 

An  electric  field  is  uniform  if  its  intensity  is  the  same  at  every 
point.  Since  E  is  a  vector,  this  condition  necessitates  a  constant 
direction  as  well  as  a  constant  magnitude.  In  most  cases  E 
varies  from  point  to  point.  Examples  of  uniform  and  other  elec- 
tric fields,  as  well  as  of  the  superposition  of  electric  fields,  will 
be  given  below. 

13.  Electric  Displacement  or  Induction.      Electrisation.      The 

physical  nature  of  every  electric  quantity  is  at  present  unknown. 
Many  phenomena,  however,  support  the  hypothesis  that  c  is  an 
elastic  permittivity  (i.  c.,  the  reciprocal  of  an  elastic  modulus) 
and  that  E  is  an  elastic  stress.  For  the  sake  of  constructing  a 
mechanical  conception  of  the  electric  field  we  shall  provisionally 
assume  c  and  E  to  be  a  permittivity  and  a  stress,  respectively. 
The  so-called  permittivity  c  will  then  be  the  actual  permittivity 
of  the  aether  or  aether  entangled  in  matter  for  the  (unknown) 
kind  of  strain  concerned. 

Now,  in  the  case  of  ordinary  elastic  substances  subjected  to 
slight  mechanical  strains  we  have,  very  approximately,  the  rela- 
tion (Hooke's  law) :  strain!  sir  ess  =  1 1  modulus  =  permittivity,  or 
strain  =  permittivity  x  stress.  If  then  c  is  a  permittivity  of  a  cer- 
tain type  and  E  a  stress  of  the  corresponding  type,  their  product 
cE  must  measure  the  corresponding  strain  or  displacement  of  the 
dielectric. 

Whether  this  conception  is  correct  or  not,  the  product  cE  is 
called  the  electric  displacement  (also  the  electric  induction),  and  is 

denoted  by  D.     That  is 

D=cE  (3) 


12  ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

E  being  a  vector,  and  c  being  the  same  for  every  direction  of 
the  intensity,  since  isotropic  substances  only  are  to  be  considered 
here,  D  is  a  vector  with  the  same  direction  as  that  of  E.  When 
c  and  E  are  expressed  in  RES  units,  D  is  said  to  be  expressed  in 
the  RES  unit  displacement  (or  induction). 

A  substance  in  which  there  is  electric  displacement  is  also  said 
to  be  in  a  state  of  electrisation,  or  to  be  electrised.  If  the  dis- 
placement and  permittivity  are  uniform  throughout,  the  electrisa- 
tion is  said  to  be  uniform. 

14.  Mechanical  Conception  of  the  Electric  Field.  A  definite 
conception  of  the  electric  field  based  on  the  assumptions  made 
above  will  now  be  given.  According  to  this  conception  (which 
leads  to  results  by  no  means  wholly  consistent,  however)  the 
aether  is  the  simplest  possible  kind  of  dielectric  and  is  composed 
of  two  kinds  of  minute,  incompressible,  elastic  cells,  called 


a.  No  electric  displacement  b.  Electric  displacement 

directed  to  left 
Fig.    1. 

positive  and  negative  cells,  respectively,  so  arranged  (in  rows), 
Fig.  I,  a,  that  only  unlike  kinds  are  in  contact,  and  that  no  slip 
between  adjacent  cells  is  possible. 

When  the  aether  supports  an  electric  field,  the  cells  remain  un- 
changed in  volume,  but  their  shapes  are  distorted  and  their  centers 
of  volume  displaced,  Fig.  I,  by  the  centers  of  the  positive  cells  in 
the  direction  of  the  electric  intensity,  and  the  centers  of  the 
negative  cells  in  the  opposite  direction.  The  electric  displace- 
ment is  measured  by  the  relative  linear  displacement  of  the 
centers  of  volume  of  the  cells  of  a  positive  row  with  reference 
to  the  centers  of  volume  of  the  adjacent  negative  rows  divided 


GENERAL  ELECTROSTATIC  THEORY.         13 

by  the  distance  between  two  adjacent  rows.  The  electric  inten- 
sity is  the  force  per  unit  area  in  the  direction  of  D  acting  upon 
the  positive  cells,  or  the  force  per  unit  area  in  the  opposite  direc- 
tion to  that  of  D  acting  upon  the  negative  cells,  in  any  plane 
passing  through  the  direction  of  D.  For  small  displacements, 
the  displacement  and  intensity  so  measured  will  be  proportional, 
as  required  by  (3)  in  all  cases.  The  total  mechanical  force  acting 
upon  the  whole  substance  within  any  element  of  volume  is  zero. 

From  what  precedes  and  from  the  nature  of  the  distortion  as 
shown  in  the  figure,  it  is  clear  that  there  is  a  tension  in  the 
aether  parallel  to  the  intensity,  and  a  pressure  in  all  directions 
normal  to  the  intensity.  That  this  deduction  from  our  mechanical 
conception  is  consistent  with  fact  is  demonstrated  in  §  §  40-4 1 . 

When  the  dielectric,  instead  of  free  aether,  is  a  molecular  sub- 
stance permeated  by  aether,  the  same  general  conception  is  use- 
ful. Like  the  aether  which  permeates  the  matter,  its  molecules 
may  be  thought  of  as  composed  each  of  two  constituents,  positive 
and  negative  atoms,  or  atomic  groups,  or  corpuscles  (Chapter 
IX.),  which  suffer  a  displacement  similar  and  in  addition  to  that 
of  the  aether  cells  entangled  among  them.  However  this  may 
be,  the  permittivity  of  all  molecular  substances  yet  investigated 
is  greater  than  that  of  free  aether.  Thus,  in  ordinary  matter  a 
greater  displacement  than  in  free  aether  accompanies  a  given 
intensity. 

In  perfect  insulators,  according  to  our  conception,  the  cells 
cannot  slip  over  one  another,  and  thus  elastic  displacement  only 
can  accompany  electric  intensity.  In  an  imperfect  insulator  the 
cells  can  slip  only  with  extreme  slowness,  and  more  slowly  the 
more  highly  insulating  the  substance.  In  a  conductor  electric 
stress  can  exist  only  temporarily  (unless  an  impressed  electro- 
motive force,  Chapter  VIII.,  is  continuously  acting),  and  is 
always  accompanied  by  rapid  slip.  That  the  substance  of  a  con- 
ductor cannot  support  electric  displacement  in  a  static  field  will 
be  shown  in  §  15.  The  mechanical  conception  of  electric  con- 
duction will  receive  further  consideration  later  on  (Chapter  IX.). 


14  ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

15.  Electric  Displacement  and  Intensity  Zero  within  a  Conduc- 
tor in  a  Static  Field.     We  may  now  restate  (4),  §  4,  as  a  corol- 
lary of  (3),  §  4  :  A  static  field  cannot  exist  within  the  substance 
of  a  conductor.      For   the  fields  within   and  without  a  hollow 
closed   conductor   are   absolutely  independent   of   one  another, 
however  thin  the  conducting  shell.      Hence  they  cannot  be  con- 
nected by  an  electric  field  or  electrically  strained  medium,  and 
the  whole  substance  of  a  conductor,  except  an  extremely  thin 
surface  layer,  is  without  electrical  significance  (in  a  static  field). 
Thus  the  electric  intensity  and  displacement  in  the  outer  region 
terminate  at  the  outer  surface  of  the  conductor,  and  the  electric 
intensity  and  displacement  of  the  inner  region  (if  the  conductor 
is  hollow  and  encloses  insulated  electrified  bodies)  terminate  at 
the  inner  surface. 

16.  Lines  and  Tubes  of  Intensity,  Displacement,  etc.     A  line 
so  drawn  in  an  electric  field  as  to  have  at  every  point  along  its 
length  the  direction  of  the  electric  intensity  (electric  force),  elec- 
trisation, or  displacement  (induction)  is   called  a  line  of  intensity 
(force),  electrisation,  or  displacement  (induction^. 

A  tubular  surface  the  elements  of  which  consist  wholly  of 
lines  of  intensity  or  induction  (etc.)  is  called  a  tube  of  intensity 
or  induction  (etc.). 

The  strength  of  a  tube  of  induction  or  displacement  is  defined 
in  §  23. 

17.  Voltage,  Electromotive  Force,  and  Difference  of  Potential. 
The  work  done  by  the  electric  field  in  carrying  an  indefinitely 
small  body  with  electric  charge  q  along  an  element  dL  (Fig.  2) 
of  a  path  L  between  two  points  /\  and  P2  of  an  electric  field,  if 
dL  makes  an  angle  6  with  the  electric  intensity  E,  is 


(4) 

and  the  total  work  done  in  carrying  q  along  L  from  Pl  to  P2  is 

(5) 


GENERAL  ELECTROSTATIC  THEORY.         15 

the  integral  being  taken  from  Pl  to  P.2.  To  carry  q  from  P2  to 
Pl  along  the  same  path  would  of  course  require  the  expenditure 
of  the  same  amount  of  work  against  the  field  by  an  outside  agent. 

In  the  same  way  the  work 
done  by  the  field  in  carrying  the 
body  with  charge  q  from  Pl  to 
P2  along  another  path  L'  is 

W  =  q$Ef  costf'  dL'. 

If  the  electric  field  is  a  static   PI 
field,  W=  W  ,  and  therefore 

f£cos  QdL  =  f£f  cos  Q'dL*. 


For  if  the  work  done  along  any  path  L  were  greater  than  that 
done  along  any  other  path  L'  ,  a  positive  amount  of  work, 
IV  —  W  t  would  be  done  on  the  charged  body  by  the  field  dur- 
ing each  completion  of  a  circuit  from  Pl  to  P2  along  L  and  back 
along  L1  ,  and  yet  the  energy  of  the  field  would  remain  unaltered. 
Since  this  is  inconsistent  with  the  principle  of  the  conservation 
of  energy,  jpjS  cos  6  dL  is  the  same  for  every  path  between  two 
given  points  in  an  electrostatic  field. 

The  line  integral  of  the  electric  intensity,  §  E  cos  6  dL  =  Wjq, 
along  a  path  L  from  P1  to  P2  is  called  the  electromotive  force 
(e.m.f.)  or  voltage  along  the  path  L  from  Pl  to  P2.  When,  as  in 
the  case  just  considered,  this  quantity  is  the  same  for  every  path 
from  Pl  to  Pv  it  is  called  also  the  difference  of  potential  between 
P1  and  P2,  or  the/0//  of  potential  from  Pl  to  Pv 

Since  a  voltage  is  a  quantity  of  work  divided  by  a  charge,  it  is 
evidently  not  a  vector. 

When  Wis  expressed  in  ergs,  and  q  in  the  RES  unit  charge, 
or  when  E  is  expressed  in  the  RES  unit  intensity,  and  L  in  cm., 
the  voltage  (  =  Wjq  =  \  E  cos  0  dL)  is  said  to  be  expressed  in 
the  RES  unit  voltage.  In  magnitude,  the  voltage  between  two 
points  is  equal  to  the  work  done  in  carrying  unit  charge  from 


l6  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

one  point  to  the  other  along  the  given  path,  or  any  path  if  the 
voltage  is  a  potential  difference. 

18.  Potential.     Equipotential  Surfaces.       The  fall  of  potential 
from  a  given  point  P  to  any  point  at  an  infinite  distance  from  all 
electrified  bodies  is  called  the  electric  potential  at  P. 

This  term  is  also  commonly  applied  to  the  fall  of  potential  from 
P  to  any  point  of  the  earth.  That  the  two  definitions  are  not 
identical  will  be  shown  in  §  6,  Chapter  II. 

The  symbol  V  will  be  used  to  denote  the  potential  at  a  point 
P.  In  conformity  with  this  notation,  the  fall  of  potential  from  a 
point  Pl  to  a  point  P2  will  be  written  Vl  —  V2,  V12,  or,  where  there 
is  no  danger  of  confusion,  simply  V. 

A  surface  which  is  everywhere  normal  to  the  electric  intensity, 
and  between  any  two  points  of  which  there  is  therefore  no  voltage, 
is  called  an  equipotential  surface,  or  simply  an  equipotential  .  It  is 
clear  that  an  equipotential  surface  is  always  a  closed  surface  or 
else  (in  certain  ideal  fields)  an  infinite  plane. 

19.  Electric  Intensity  in  a  Static  Field  the  Space  Rate  of  Dimi- 
nution of  Potential.  —  For  the  voltage  from  P^  to  P2  we  have 


Vl  -  V2  =  /£cos  BdL  =  f£L  dL 

by  writing  EL  for  E  cos  0,  the  component  of  electric  intensity  in 
the  direction  of  dL.  That  is,  the  potential  of  Pl  exceeds  that  at  P2 
by  §  EL  dL  from  Pl  to  P2  ;  or  the  diminution  of  potential  from  Pl 
to  P2  is  j/£j  dL  from  Pl  to  P2.  If  the  two  points  are  taken  an 
infinitesimal  distance  dL  apart,  the  diminution  of  potential  along 
dL  becomes  —  dVy  and  the  integral  becomes  simply  EL  dL.  Thus 

we  have 

-  dV=  EL  dL 
and  therefore 

EL  =  -  dVldL  (6) 


That  is,  the  component  of  electric  intensity  in  any  direction  is  the 
space  rate  of  diminution  of  the  electric  potential  in  that  direction. 


GENERAL  ELECTROSTATIC  THEORY.         I/ 

V  obviously  diminishes  most  rapidly  along  a  line  of  intensity, 
and  not  at  all  along  a  line  in  an  equipotential  surface. 

20.  Electric  Field  Mapped  out  by  a  System  of  Equipotentials. 

If  a  line  of  intensity  is  denoted  by  Nt  the  last  equation  gives 

EN=E=  -dVjdN. 

From  this  it  follows  that  an  electric  field  can  be  completely 
mapped  out  by  a  system  of  equipotential  surfaces  so  drawn  that 
the  voltage  between  successive  surfaces  is  constant.  For  the 
direction  of  the  intensity  at  any  point  is  that  of  the  normal  to  the 
equipotential  passing  through  the  point ;  and  its  magnitude  is, 
by  the  above  equation,  proportional  to  the  number  of  successive 
equipotential  surfaces  crossed  at  the  point  per  unit  length  by 
this  normal  or  line  of  intensity.  Maxwell's  method  of  drawing 
such  an  equipotential  system  is  described  in  §§7,  n,  13,  14,  II. 

21.  Electric  Flux.     Let  dS,  Fig.  3,  denote  an  element  of  area 
at  any  point  of  an  electric  field  where  the  displacement  is  D,  and 
let  the  angle  between  D  and  the  normal  N  to  dS  be  denoted  by 
0.     The  product  of  dS  into 

the  component  of  D  normal 
to  dS,  that  is,  D  cos  6  dS, 
is  called  the  electric  flux 
across  dS. 

To  obtain  the  electric  flux, 
II,  across  an  extended  sur- 
face S,  over  which  D  may 
vary  in  any  manner,  the  in- 
tegral of  D  cos  6  dS  must 
be  taken  over  the  whole  sur-  Re-  3- 

face.     Thus 

n  =  fl>  cos  0  dS  (7) 

22.  Gauss's  Theorem:  The    electric  flux    outward  across  any 
closed  surface  5  so  drawn  as  to  enclose  a  total  charge  q  is  equal 
to  q. 


1 8  ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

The  theorem  will  first  be  established  for  the  case  in  which  all 
space  is  filled  up  with  a  single  homogeneous  isotropic  dielectric 
with  permittivity  c  (or  with  any  number  of  isotropic  dielectrics  all 
of  which  have  the  same  permittivity  c). 


Fig.  4.        . 

Consider  first  the  field  about  a  charge  q  concentrated  at  P, 
Fig.  4,  any  point  within  5,  a  closed  surface  of  any  shape.  For 
the  magnitude  of  the  displacement,  D,  at  any  element  of  area 
dS,  distant  L  from  P,  we  have  from  (i),  (2),  and  (3) 

D=cE=  c(ql4  TT  cL2)  =  qj^irL2 

In  direction,  D  and  E  are  evidently  radial  from  P  (or  to  P  if 
q  is  negative). 

For  the  flux  across  dS  we  have  therefore 

dH  =  D  cos  6  dS  =  q  dS  cos  0/47rL2  =  q  dS'l^D  =  ql^rr  •  da> 

where  dSf  =  dS  cos  6  is  the  projection  of  dS  normal  to  L,  and 
d<£>  =  dSf  JL?  is  the  elementary  solid  angle  subtended  at  P  by  dS 
and  dSf ',  that  is,  the  angle  of  the  elementary  conical  tube  of  in- 
duction cutting  out  the  area  dS. 

If  the  surface  is  folded,  so  that  some  of  the  tubes  cut  it  more 
than  once,  as  the  tube  of  angle  da  which  cuts  out  the  areas  dSit 


GENERAL  ELECTROSTATIC  THEORY.         19 

dS2,  •  •  - ,  dSb  in  the  figure,  each  of  these  tubes  must  obviously 
cut  it  an  odd  number  of  times.  And  since  the  angle  dw  of  the 
cone  is  the  same  for  all  the  elements  dSv  dS2,  etc.,  the  magnitude 
of  the  flux  across  each  will  be  the  same,  viz.,  gj^Tr-dw,  but  the 
flux  will  be  outward  (positive)  across  all  the  elements  with  odd 
numbers,  and  inward  (negative)  across  all  the  elements  with  even 
numbers.  Thus  all  the  elements  except  one,  across  which  the 
flux  is  positive  or  outward,  cut  one  another  out  in  pairs,  leaving 
the  total  flux  outward  through  the  tube  equal,  as  for  a  tube  of 
the  same  angle  cutting  the  surface  but  once,  to  ^/4?r  •  dot. 
The  outward  flux  across  the  complete  surface  is  therefore 

II  =  fdft  =  qjqir  fdto  =  q  (8) 

since  the  whole  solid  angle,    f  dco,  subtended  by  any  closed  sur- 
face at  a  point  within  it  is  477-. 

This  result  is  independent  of  the  position  of  P  within  S\  hence, 
by  the  principle  of  superposition,  it  must  hold  for  charges  dis- 
tributed in  any  manner  within  S,  q  denoting  now  the  total  (alge- 
braic) charge  within.  The  validity  of  the  theorem  for  all  isotropic 
electrostatic  fields  will  be  established  later  (§§  29,  I.  and  i,  IV.). 

23.  The  Strength  of  a  Tube  of  Induction.  From  (8)  it  fol- 
lows that  the  flux  across  every  cross-section  of  a  given  tube  of 
induction  is  the  same.  For  by  the  definition  of  a  tube  there  is 


Fig.  5. 


no  flux  across  any  part  of  its  sides ;  and  since  in  the  space  en- 
closed within  the  sides  and  two  diaphragms  6\  and  Sv  Fig.  5, 
there  is  no  electric  charge,  the  flux  which  enters  this  region 


20  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

across  vSL  must  equal  that  which  leaves  across  S2.  Thus  there 
is  an  analogy  between  the  electric  flux  and  the  flux  of  an  incom- 
pressible fluid. 

The  strength  of  a  tube  of  induction  is  defined  as  the  magnitude 
of  the  flux  across  any  diaphragm  of  the  tube.  A  unit  tube  is  a 
tube  whose  strength  is  unity. 

24.  Electric  Charge  and  Discontinuity  of  Electric  Flux.  With 
the  exception  of  closed  tubes  of  induction  (Chapter  VI.),  all 
tubes  in  a  static  field  emanate  from  positively  charged  bodies  and 
terminate  upon  negatively  charged  bodies.  To  prove  this  state- 
ment, consider  two  electrified  bodies  (there  cannot  be  less  than 
two)  alone  in  the  field,  there  being  no  charges  upon  other 
bodies.  If  one  possesses  the  charge  -j-  q,  the  other  possesses  the 
charge  —  q  (  §  6).  The  total  electric  flux  outward  across  any  closed 
surface  surrounding  -f  q  is  q,  and  the  total  inward  flux  across 
any  closed  surface  surrounding  —  q  is  q  ;  or  the  total  flux  across 
any  closed  surface  separating  the  charge  -f-  q  from  the  charge 
—  q  is  equal  to  q  in  magnitude,  and  in  direction  is  from  -f  q 
toward  —  q.  That  is,  all  the  tubes  emanate  from  the  body 
with  charge  -f  q  and  terminate  upon  that  with  charge  —  q,  the 
total  strength  of  all  the  tubes  being  q. 

Exactly  the  same  mode  of  reasoning  may  be  applied  to  a  single 
tube  of  induction.  The  strength  of  a  tube  is  thus  equal  to  the 
magnitude  of  the  positive  charge  at  one  end  or  to  the  magnitude 
of  the  negative  charge  at  the  other.  The  whole  electric  field 
indeed  may  be  regarded  as  a  single  tube  of  induction  passing 
from  one  charge  to  the  other. 

Thus  the  electric  charge  resides  only  where  the  displacement 
is  discontinuous,  and  is  measured  by  the  amount  of  this  discon- 
tinuity. In  fact  Gauss's  theorem  simply  states  the  identity  of  an 
electric  charge  and  the  flux  from  the  charge,  or  rather  the  dis- 
continuity of  the  flux  at  the  charge. 

Rational  Units.  The  system  of  units  here  adopted  is  called 
rational  for  the  reason  that  it  makes  the  flux  from  a  charge  equal 


GENERAL  ELECTROSTATIC  THEORY.         21 

to  the  charge  numerically,  as  it  is  dimensionally,  instead  of  to  4?r 
X  the  charge,  as  in  the  common  systems,  and,  as  a  consequence, 
does  away  with  the  factor  TT  except  in  the  case  of  spherical  or 
circular  distributions,  where  it  would  naturally  occur. 

25.  Electric  Field  Mapped  Out  by  Tubes  of  Induction.     In  the 

elementary  tube  T,  Fig.  6,  let  the  diaphragms  dSv  dS2,  be  drawn 


Fig.  6. 


at  right  angles  to  the  axis  of  the  tube.  Then  we  have,  by  §23, 
DldSl  —  D2dS2,  whence 

DJDt  =  dSJdS,  =  EJE,  (9) 

Thus  the  intensity  and  induction  at  every  point  along  a  narrow 
tube  are  inversely  proportional  to  its  right  cross-section  at  the 
point.  Since  therefore  the  magnitude  of  the  right  cross-section 
of  a  tube  at  a  point  indicates  the  magnitude  of  the  induction  and 
intensity,  and  the  direction  of  the  tube  the  direction  of  these 
quantities,  an  electric  field  may  be  completely  mapped  out  by 
drawing  a  system  of  tubes,  all  of  the  same  strength,  filling  the 
field.  Maxwell's  method  of  drawing  such  a  system  of  tubes  will 
be  explained  in  §§  7,  n,  13-14,  II. 

26.  The  Surface  of  a  Conductor  in  a  Static  Field  is  an  Equipo- 
tential  Surface.     For,  since  in  a  static  field  there  is  no  electric 
intensity    within    the    substance    of    a    conductor,    the   voltage 

CE  cos  OdL  is  zero  along  any  line  drawn  wholly  through  the 
substance  of  a  conductor  and  connecting  any  two  points  of  its 
surface  (and  therefore  along  any  other  line  connecting  the  two 
points,  since  the  field  is  static). 

27.  Equipotential  Region.      If  in  the  region  on  one  side  of  a 
given  equipotential  surface  there  is  no  electric  charge,  the  elec- 


22  ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

trie  induction  and  intensity  in  this  region  are  also  zero,  and  all 
parts  of  it  are  therefore  at  the  same  potential  as  that  of  the 
equipotential  surface.  For  all  the  tubes  which  cross  an  equipo- 
tential  surface  cross  it  normally  and  but  once ;  and  in  the  region 
considered  there  is  no  electric  charge  with  which  such  tubes 
could  originate  or  terminate.  Hence  there  are  no  tubes  in  the 
region,  by  Gauss's  theorem,  and  no  voltage. 

That  the  space  containing  the  substance  of  a  conductor,  or  the 
space  included  within  a  hollow  closed  conductor,  is  an  equipo- 
tential region,  has  already  been  established. 

28.  In  a  Static  Field  a  Conductor  may  be  Replaced  by  a  Die- 
lectric of  any  Permittivity.     Since  there  is  no  electric  field   in  a 
region    without   charge    bounded    by    an    equipotential    surface 
(charged  or  uncharged),  the  substance  filling  this  region  may  be 
replaced  by  any  other  substance,  with  its  surface  charged  in  the 
same  manner  as  that  of  the  substance  replaced,  without  in  any 
way  affecting  the  electric  field.     Thus  it  is  extremely  convenient 
for  the  purpose  of  solving  many  electric  problems,  to  imagine 
the  substance  of  an  electrified  conductor  replaced  by  a  dielectric 
of  the  same  permittivity  as  that  of  the  surrounding  medium,  with 
its  surface  coincident  with  that  of  the  conductor  and  charged  in 
the  same  manner.     This  is  in  order  to  apply  the  law  of  inverse 
squares,  which  can  be  done  only  when  all  space  contains  the 
same  dielectric  of  uniform   permittivity.      Extensive  use  will  be 
made  of  this  principle  in  what  follows  and  it  will  be  generalised 
in  Chapter  IV. 

29.  Gauss's  Theorem  Valid  for  a  Finite  Region  and  for  a  Field 
Containing  or  Bounded  by  Conductors.     As  an  immediate  corollary 
of  what   precedes,    it    follows    that    Gauss's    theorem    is    valid 
throughout  an  infinite  electric  field  containing  a  homogeneous 
isotropic  dielectric  and  any  number  of  conductors.     And  as  an 
immediate  corollary  of  this   last   proposition  and   §  4,  it  follows 
that  the   theorem   is   valid  throughout   any  finite  electric  field 


GENERAL  ELECTROSTATIC  THEORY.         23 

bounded  by  conductors,  and  throughout  a  finite  portion  of  any 
electric  field,  provided  that  this  finite  field  or  portion  of  a  field 
contains  only  a  single  homogeneous  isotropic  dielectric  and  con- 
ductors. The  validity  of  the  theorem  is  still  further  extended 
in  Chapter  IV.  i 

30.  Electric  Surface  and  Volume  Density.  Convergence  and 
Divergence  of  a  Vector.  The  electric  surface  density  at  any 
point  of  a  charged  surface  is  defined  as  the  charge  per  unit  area 
at  the  point,  and  will  be  denoted  by  cr.  If  dS  is  an  element  of 
area  at  the  point  and  dq  its  charge, 


(10) 

The  outward  flux  across  any  surface  enclosing  dq  and  no 
other  electric  charges  is  dTL  =  dq  =  (?dS,  by  Gauss's  theorem 
(not  yet  proved  for  this  case,  since  the  surface  encloses,  in 
general,  two  dielectrics).  Let'  such  a  surface  be  formed  by 
a  right  cylinder  of  infinitesimal  length  drawn  through  the 


boundary  of  dS  and  closed  up  by  two  planes  parallel  with 
dSy  one  on  each  side,  Fig.  7.  The  lateral  area  of  this  cylinder  is 
negligible  in  comparison  with  that  of  the  ends,  so  that  the  out- 
ward flux  across  the  total  surface  is  equal  to  the  flux  across  the 
ends.  Therefore,  if  D^  and  £>2  are  the  displacements  on  the  two 


24  ELEMEN1S   OF   ELECTROMAGNETIC    THEORY. 

sides  of  dS,  and  0l  and  02  the  angles  they  make  with  the  normals 
drawn  from  dS, 


=  (Dl  cos  0l  +  D2  cos 
whence 

<r  =  Dl  cos  0l  -f  -£>2  cos  02  =  c^  cos  ^  +  cf2  cos  02    (i  i) 

if  cl  and  r2  denote  the  permittivities  of  the  media  on  the  two  sides 
ofdS. 

If  the  charged  surface  is  that  of  a  conductor  in  a  static  field, 
the  displacement,  D,  on  one  side  is  normal  to  the  surface,  and 
on  the  other  side  is  zero  ;  so  that  in  this  case  (i  i)  becomes 

o-  =  D  =  e£  (12) 

which  might  have  been   written   down   at   once  from    Gauss's 
theorem,  already  established  for  this  case. 

The  electric  volume  density  at  any  point  of  an  electrified  volume 
is  defined  as  the  charge  per  unit  volume  at  the  point,  and  will  be 
denoted  by  p.  If  dq  is  the  charge  in  the  element  of  volume  dr 

at  the  point, 

(13) 


The  electric  flux  outward  from  dq  through  the  surface  of  dr  is 
dq  =  pdr,  whence 

p  =  dqjdr  =  dft/dr  =  div  D  (14) 

The  symbol  div  D  is  an  abbreviation  for  the  divergence  of  D, 
which  is  another  name  for  dUjdr,  the  outward  flux  of  the 
vector  D  per  unit  volume,  or,  in  magnitude,  the  amount  of 
the  flux  leaving  unit  volume  through  part  of  its  surface  minus 
the  amount  entering  the  same  volume  through  the  rest  of  its 
surface. 

If  p  is  negative,  div  D  is  also  negative,  or  the  flux  is,  on  the 
whole,  directed  into  dr.  To  the  negative  of  the  divergence  the 
term  convergence  is  applied.  It  is  written  conv.  Hence 

—  p  =  —  div  D  =  conv  D  (15) 


GENERAL  ELECTROSTATIC  THEORY. 


The  convergence  or  divergence  of  any  other  vector  is  simi- 
larly defined  as  the  inward  or  outward  flux  of  the  vector  per 
unit  volume  at  the  given  point. 

An  insulator  may  possess  both  volume  density  and  surface 
density  of  electrification,  but  the  charge  of  a  conductor  in  a  static 
field  resides,  as  has  been  already  shown,  on  the  surface  only. 
This  statement  must  not  be  taken  too  literally,  however,  as  the 
molecular  structure  of  matter  makes  it  necessary  that  the  dis- 
placement should  terminate  upon  the  atoms  of  a  surface  layer, 
although  this  layer  is  extremely  thin. 

31.  Cartesian  Expression  for  the  Divergence  and  Convergence 
of  a  Vector.  The  Equations  of  Poisson  and  Laplace.  First  we 
shall  obtain  the  expression  for  the  divergence  of  the  vector  D. 
Let  the  components  of  D  at  the  point  whose  coordinates  are  xy 
y,  2t  parallel  to  the  rectangular  axes  X,  Y,  Z,  be  Dv  Dv  Z>3, 
respectively,  Fig.  8.  Consider  the  elementary  parallelepiped 
whose  edges  are  parallel  to  the  coordinate  axes  and  have  the 
infinitesimal  lengths  dx,  dy, 
dz,  the  coordinates  of  the 
corner  nearest  the  origin  of 
coordinates  being  x,  yy  z. 

The  flux  into  the  parallel- 
epiped through  the  face  13 
is  D2  dxdz  (or  the  flux  out 
across  the  face  13  is  —  D2 
dxdz),  and  that  out  through 
the  opposite  face,  57,  is  (D2 
+  dDJdy  dy)  dxdz.  Hence 
the  resultant  outward  flux 
across  the  two  faces  parallel 
to  the  JfZplane  is  (D^+dDJdydy^dxdz—D^dxdz^dDjdy  dxdydz. 

In  exactly  the  same  way  the  resultant  outward  flux  across  the 
two  faces  26  and  35  is  dDJdx  dxdydz,  and  that  across  the  faces 
46  and  37,  dDJdz  dxdydz. 


Fig.  8. 


26  ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

Hence  the  total  flux  outward  from  the  parallelepiped  is 


=  (dDJdx  +  dDJdy  +  dDJdz)  dxdydz 

=  (dDJdx  +  dDJdy  +  dDJdz)  dr 
and 

p  =  div  D  =  —  conv  D  =  d  H/dr 

(16) 
=  ^/^r  +  aizy^  +  dDJdz 

Similarly,  for  any  other  vector,  as  £,  we  have 

div  E  =  -  conv  E  =  dEJdx  +  dEJdy  +  dEJdz 
If  £  is  constant  (independent  of  xt  y,  z),  we  have,  since  D  =  cEy 
div  D  --=  dDJdx  +  dDJdy  + 


(17) 
=  c(dEJdx  +  dEJdy  +  dEJdz)  =  c  div  E 


Equation  (  1  7)  may  be  written 
p  =  div  D  = 


=  -  djdx(cdVld*)  -  d/dy  (cdVjdy)  -  djdz  (cdVjdz) 
If  c  is  independent  of  the  coordinates,  this  equation  becomes 
p  =  div  D  =  -c(d*Vjd^  +  d*Vjdy*  +  d2Vldz2)         (19) 

(18)  and  (19)  are  the  equations  of  Poisson.     When  p  =  o,  the 
equations  become 

(cdVldx)  +  <//*#/  (cdVjdy}  +  djdz  (cdV/dz)  =  o     (20) 


and 

d*  Vjdx2  +  d2  Vjdy*  +  d2  Vj  dz2  =  o  (21) 

which  are  the  equations  of  Laplace. 

32.  The  Equilibrium  of  Superposed  Electric  Fields,  (i)  If  in 
each  of  any  number  of  electric  fields  separately  each  of  a  given 
system  of  surfaces  of  fixed  configuration  is  an  equipotential,  then 
in  the  electric  field  resulting  from  the  geometric  superposition  of 
these  fields  each  surface  will  remain  an  equipotential.  For  since 
in  each  field  separately  the  tubes  meet  the  surfaces  normally,  by 


GENERAL    ELECTROSTATIC    THEORY.  2/ 

definition   of  an   equipotential,   the   tubes   in   the   geometrically 
obtained  resultant  field  will  also  meet  the  surfaces  normally. 

(2)  That  the  superposition  of  any  number  of  distributions  of 
electric  charges  in  or  upon  insulators  gives  a  resultant  distribution 
of  charges  in  equilibrium  is  evident  from  the  definition  of  an  insu- 
lator.    That  the  resultant  field  (obtained  by  geometrical  super- 
position) connected  with  these  charges  is  in  equilibrium,  and  that 
this  is  the  only  possible  resultant  field  in  equilibrium,  follows 
from  §12. 

(3)  If  each  or  any  of  the  equipotentials  of  (i)  encloses  no 
charges,  then  it  encloses  no  field,  and  it  is  immaterial  so  far  as 
the  external  (/.  e.y  the  only)  field  is  concerned  whether  the  sub- 
stance within  this  surface  is  an  insulator  or  a  conductor  (§28). 
If  the  field  is  in  equilibrium  in  the  one  case,  it  will  be  in  equi- 
librium in  the  other.      Hence  we  may  state  that  if  each  of  any 
number  of  electric  fields  surrounding  or  bounding  a  given  system 
of  conductors  with  fixed  configuration  is  separately  in  equilibrium, 
then  the  electric  field  resulting  from  their  geometric  superposition 
will  also  be  in  equilibrium,  and  will  be  the  only  possible  resultant 
field  in  equilibrium  (i.  e.y  static).     The  last  statement  is  proved 
again  in  §  46. 

33.  The  Superposition  of  Voltages  and  Potentials.  If  the  vol- 
tage from  any  point  P  to  any  other  point  P  is  Vl  when  the  field 
surrounding  the  points  is  a  given  field  Av  V2  when  the  field  is 

Av  '  •  '  >  Vn  when  the  field  is  An>  tnen  tne  voltage  from  P  to  P 
when  all  the  fields  are  superposed  is 

V~     Vl+     ^  +     •••+     Vn  (22) 

For,  all  the  integrals  being  taken  along  the  same  path  L  (which 
may  be  any  path  from  P  to  /*),  we  have,  in  the  notation  of 
§§  17-19, 


=          cos    i>     i  =        *  cos    2<>  '  '  '  ,     n  =          cos 
and  V=f£cos0dL 


28  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

By  the  principle  of  superposition  of  electric  intensities 

E  cos  0  =  El  cos  6l  +  E2  cos  02  +  •  •  •  -f  £H  cos  6 
Hence 

F=     £  cos  BdL  =  V,  +  F2  +  -  .  .  +  F 


which  is  identical  with  (22). 

If  /*  is  any  point  in  the  region  of  zero  potential,  Fand  Vlt 
V2  ,.•••,  Vn  denote  the  resultant  and-  component  potentials,  re- 
spectively at  P. 

34.  Voltages  and  Charges  Proportional.     It  is  clear  from  §§32 
and  33  that  when  the  intensity  at  every  point,  and  therefore  the 
voltage  between  every  two  points,   of  a  static   electric  field  is 
altered  in  any  ratio,  the  resulting  electric  field  will  be  in  equi- 
librium, and  the  electric  surface  or  volume  density  at  every  ele- 
ment of  charged  surface  or  volume  will  be  altered  in  the  same 
ratio,  and  vice  versa.     The  original  field  has  simply  been  super- 
posed on  itself  a  given  number  of  times. 

35.  Capacity  of  an  Electrical  System.     Permittance  of  a  Dielec- 
tric.     S  is  Proportional  to  c.      In  an  electric  field  terminated  by 
two  conductors  A  and  B  all  the  tubes  emanate  from  one  of  the 
conductors  and  terminate  upon  the  other,  so  that  the  charges  of 
A  and  B  are  equal  and  opposite  whatever  their  common  magni- 
tude, q.     This  relation  still  holds  when  any  number  of  other  con- 
ductors, uncharged  except  by  induction,  are  in  the  field,  the  tubes 
connecting  A  and  B  simply  being  rendered  discontinuous  at  the 
surfaces  of  these  conductors  (§  42).     By  the  last  article,  if  the 
voltage  V12  between  A  and  B  is  altered  in  any  ratio,  q  will  be 
altered  in  the  same  ratio,  and  vice  versa.     That  is 


(23) 

where  5  is  a  constant,  called  the  capacity  of  the  system  AB,  or 
much  better,  the  capacity  or  permittance  of  the  dielectric  bounded 
by  A  and  B. 


GENERAL    ELECTROSTATIC    THEORY.  29 

The  above  equation  may  be  written 

lfEdL        ,  (24) 

dS  being  the  element  of  area  of  any  equipotential  surface  (neces- 
sarily closed  around  one  of  the  conductors  or  else  extending  to 
infinity),  and  dL  the  element  of  length  of  any  line  of  intensity,  the 
integrals  extending  over  the  whole  surface  and  along  the  whole 
length  of  the  line,  respectively. 

The  term  permittance  is  applied  to  5  (surface  integral  of  dis- 
placement/line integral  of  intensity)  for  the  same  reason  for 
which  the  term  permittivity  is  applied  to  c  (displacement/intensity). 

When  the  charge  and  voltage  are  expressed  in  RES  units  in 
(23)  and  (24),  5  is  said  to  be  expressed  in  the  RES  unit  capacity 
or  permittance.  The  unit  capacity  is  thus  the  capacity  of  a  sys- 
tem, or  dielectric,  upon  each  of  whose  terminating  conductors 
the  charge  is  unity  when  the  voltage  is  unity. 

Two  conductors  which  completely  bound  an  electric  field,  like 
the  system  AJ5,  are  called,  with  the  intervening  dielectric,  an 
electric  condenser  or  leyden.  These  terms  are  commonly  applied, 
however,  only  when  the  conductors  are  near  together,  in  which 
case  the  displacement  may  be  very  great,  or  the  electric  charge 
highly  "  condensed"  even  when  the  voltage 


is  small  (since  L  is  small).  The  term  condenser  is  also  applied 
to  a  system  in  which  nearly  all  the  tubes  of  induction  pass  from 
one  of  the  conductors  to  the  other. 

If  the  voltage  V  of  a  leyden  is  kept  constant,  and  the  permit- 
tivity c  of  its  dielectric  altered  everywhere  in  a  given  ratio,  the 
intensity  E  will  remain  constant,  but  the  displacement  D,  and 
therefore  the  charge  q,  will  be  altered  everywhere  in  the  same 
ratio.  Hence  5  is  proportional  to  c. 

Although  a  charge  of  one  sign  cannot  exist  without  the  com- 
plementary charge  of  opposite  sign,  it  is  sometimes  convenient 


30  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

to  imagine  one  of  the  charges  removed  to  an  infinite  distance, 
when  the  electric  field  within  a  finite  distance  is  connected  with 
only  a  single  electrified  body  and  conductors  with  induced 
charges.  The  intensity  and  potential  at  every  point  will  then  be 
proportional  to  the  charge  of  the  electrified  body. 

36.  Mechanical  Analogue  of  the  Relation  q  =  S1712.     If  a  spring 
(analogous  to  the  dielectric  of  a  condenser)  which  obeys  Hooke's 
law  and  has  perfect  elasticity  (c  =  D JE  =  constant)  is  stretched 
a  distance  L  (analogous  to  q)  by  a  force  F  (analogous  to  F12)  then 

L~KF  (25) 

where  K  (analogous  to  5 )  is  a  constant  depending  on  the  spring. 
A  similar  relation  of  course  exists  between  the  deformation  and 
the  forcive  in  the  case  of  any  other  perfectly  elastic  strain. 

37.  The  Electrostatic  Energy  of  a  Field  Bounded  by  Two  Con- 
ductors,    Energy  Contained  in  a  Tube  of  Displacement  Between 
Two    Equipotentials.     The    energy   contained    in    the    dielectric 
bounded  by  the  two  conductors  A  and  B  due  to  its  electric  dis- 
placement is  equal  to  the  work   done  in   creating  the   electric 
field,  or  the  work  done  against  the  electric  field  in  charging  the 
system    (provided    there  is  no  dissipation  of  energy  in  dielec- 
tric hysteresis,  §  I,  VI.).      Let  the  process  of  charging  consist  in 
carrying  successive  elements  of  charge  dq  from  B  to  A,  or  —  dq 
from  A  to  B,  or  both.     Each  time  this  is  done  A  gains  a  charge 
-f  dq  and  B  a  charge  —  dq,  and  the  work  done  in  effecting  the 
transfer,  if  the  charges  of  A  and  By  at  the  time  are  -f-  q  and  —  q 
respectively,  and  if  the  corresponding  voltage  from  A  to  B  is 

y,  is 

dW=  Vdq  (26) 

by  (5). 

If  S=  ql  V=  constant,  which  is  true  except  when  intrinsic 
displacement  (VI.)  is  present,  (26)  may  be  written 

dW=  Vdq=  \ISqdq  =  SVdV 


GENERAL  ELECTROSTATIC  THEORY.         31 

Hence  the  total  work  done  in  establishing  the  field,  or  the  total 
electrostatic  energy  of  the  field,  is 


=  i/S 

°     r 

=  S 

Jo 


The  electric  field  may  be  considered  as  a  single  tube  of  dis- 
placement connecting  A  and  _Z>,  the  strength  of  the  tube  being 
q  and  its  voltage  V.  The  energy  of  this  tube  is  then  one  half 
the  product  of  its  strength  by  its  voltage.  Or  the  field  may  be 
divided  up  into  tubes  of  displacement  in  any  manner,  and  since 
the  above  result  is  wholly  independent  of  the  shapes  of  the  tubes, 
the  energy  contained  in  each  tube  is  in  the  same  way  one  half  of 
the  product  of  its  strength  by  its  voltage.  Also,  the  energy  con- 
tained in  the  portion  of  any  tube  of  strength  q  between  two  equi- 
potential  surfaces  differing  in  potential  by  V  is  ^q  V,  whether  the 
tube  terminates  at  these  surfaces  or  not. 

In  any  case,  whether  energy  is  dissipated  or  not,  or  whether 
qjV—S  is  constant  or  not,  the  work  done  in  charging  the  con- 
denser from  a  neutral  state  to  charge  q,  or  the  work  done  in 
changing  the  strength  of  a  tube  of  displacement  from  0  to  q, 
and  its  voltage  from  0  to  V,  is 


Vdq 


(28) 


38.  Electric  Energy  Density  in  a  Dielectric.  From  §  37  it  fol- 
lows that  when  D  =  cE  (no  intrinsic  electrisation  present,  Chapter 
VI.)  the  energy  per  unit  volume  at  any  point  of  an  electric  field  is 

U-\ED=\cE*  (29) 

To  prove  this,  consider  an  elementary  tube,  of  strength  dq, 
cutting  two  equipotential  surfaces  distant  dL  apart,  the  point  con- 
sidered being  at  the  center  of  the  element  of  volume  dr  enclosed 
by  the  sides  of  the  tube  and  the  equipotentials.  If  the  right 


32  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

cross -section  of  the  tube  is  dS  at  the  point,  dr  =  dLdS.     The 
energy  contained  in  dr  is 

dW=  \dq  EdL  =  \DdS  EdL  =  \ED  dr  =  *cE2  dr 
and  the  energy  per  unit  volume  is 

U=  dWjdr  =  \ED  =  J  cE2 

which  is  identical  with  (30). 

Without  assuming  the  relation  qj  V  =  S  =  constant,  of 
c  =  Dj E  =  constant,  or  that  there  is  no  dissipation  of  energy, 
we  can  show  that  the  work  done  per  unit  volume  in  creating  a 
displacement  D  is 

U=   ^EdD  (30) 

i/O 

which  reduces  to  (29)  when  c  =  DjE  =  constant. 

For  in  the  general  case,  §  37  (28),  the  work  done  in  changing 
the  strength  of  a  tube  of  displacement  from  0  to  q  is 

W=fVdq  =  fff  EdDdLdS  =  //  EdD  dr 
which,  on  differentiating  with  respect  to  r,  gives  (30). 

39.  Electric  Tension  and  Pressure  (Preliminary).  From  the 
consideration  of  a  static  electric  field  (such  as  the  field  of  Fig.  22, 
24,  or  47),  in  which  tubes  of  induction  stretch,  in  general,  from 
a  positively  charged  body  to  another  body  negatively  charged ; 
in  which  there  is  always  a  force  of  attraction  between  the  op- 
positely charged  bodies ;  and  in  which  a  small  electrified  body 
(if  the  force  of  gravity  is  eliminated)  will  move  along  a  line  of 
intensity ;  it  follows  immediately  that  at  every  point  of  an  elec- 
tric field  there  is  a  tension  in  the  dielectric  in  a  direction  parallel 
to  the  intensity  —  the  tubes  of  induction  tending  to  contract  in 
length  indefinitely  and  to  pull  together  the  electrified  bodies  on 
which  they  end. 

It  is  clear  also  from  the  manner  in  which  the  tubes  of  induction 
spread  out  laterally  as  they  pass  from  one  of  the  bodies  to  the 


CxENERAL    ELECTROSTATIC    THEORY.  33 

other,  filling  all  space  except  as  the  field  is  bounded  by  conduc- 
tors, that  at  every  point  in  the  dielectric  there  is  a  pressure  per- 
pendicular in  every  direction  to  the  intensity  at  the  point.  Were 
the  tension  along  the  tubes  the  only  stress,  it  is  clear  that  all  the 
tubes  would  contract  in  cross-section  as  well  as  in  length  and 
stretch  straight  across  from  one  charge  to  the  other  ;  and  the 
electromotive  force  from  charge  to  charge  along  all  paths  not 
passing  through  the  region  occupied  by  these  tubes  would  be 
zero,  which  is  of  course  impossible. 

It  is  clear  also  that  the  i4ite#sity  and  pressure  at  any  point  are 
greater  the  greater  the  intensity  and  induction  at  the  point. 
These  stresses  are  referred  to  in  §  14,  and  will  receive  detailed 
consideration  in  the  next  two  articles. 

40.  Electric  Tension,  Method  I.  At  any  point  in  a  dielectric  in 
which  (29)  holds  there  is  a  tension  in  the  direction  of  the  in- 
tensity, with  magnitude  per  unit  area 


To  prove  this,  consider  a  uniform  field  which  is  terminated  at 
one  end  by  a  plane  conducting  plate  of  area  A  (necessarily)  nor- 
mal to  the  electric  field  (III,  §  2).  If  the  plate  is  moved  in  a  direc- 
tion parallel  to  the  field  an  infinitesimal  distance  dL,  the  volume 
of  the  dielectric  under  strain  terminated  by  the  plate  of  area  A  is 
increased  by  AdL  and  the  energy  by  dW  '=  AdL^ED,  the  dis- 
placement of  the  plate  being  so  small  that  E  and  D  remain  sen- 
sibly unaltered.  This  increase  in  energy  is  equal  to  the  work 
done  in  moving  the  plate  the  distance  dL  against  the  force 
normal  to  its  surface  due  to  the  tension  in  the  dielectric.  If  the 
force  per  unit  area  on  the  plate,  which  must  equal  the  tension  in 
the  direction  of  the  intensity  in  the  dielectric,  is  denoted  by  T, 
we  have  therefore 


dW=  TAdL  =  \EDAdL  ;  and  T=  i/A  dWjdL  =  \ED,  etc., 
which  is  identical  with  (31). 


34  ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

This  result  has  been  deduced  for  a  uniform  field,  but  since 
every  field  is  uniform  throughout  an  infinitesimal  volume,  the 
result  is  perfectly  general. 

The  best  form  of  apparatus  for  investigating  the  electric  tension 
experimentally  is  described  in  §§  2  and  4,  III. 

Electric  Tension,  Method  II.  The  proposition  just  established 
may  also  be  demonstrated  as  follows  :  Let  dS  be  an  element 
of  the  charged  surface  of  a  conductor,  and  let  P  and  Q  be  two 
points  indefinitely  near  the  surface,  one  without  and  the  other 
within  the  conductor  at  the  center  of  dS.  Consider  the  sub- 
stance of  the  conductor  replaced  by  a  dielectric  of  the  same  per- 
mittivity as  that  of  the  surrounding  medium  (c),  §  28.  Then 
the  electric  intensity  at  P  and  at  Q  may  be  resolved  into  two 
components,  one  which  can  be  calculated  from  the  charge  upon 
dS,  and  the  other  from  the  rest  of  the  charges  in  the  field  (or,  as 
ordinarily  expressed,  one  due  to  the  charge  on  dS,  and  the  other 
due  to  the  other  charges).  Let  the  two  components  at  P  be 
denoted  by  El  and  Ev  and  the  resultant  intensity  by  E.  By 
symmetry,  El  is  normal  to  dS,  and  there  is  an  equal  and  oppo- 
site component,  —  Ev  at  Q.  Since  E  and  El  are  both  normal 
to  dSt  Ev  their  vector  difference,  is  also  normal.  Hence 

E=El  +  E2 

The  resultant  intensity  at  Q,  inside  the  surface,  is  zero,  and 
has  the  components,  E2  normally  outward,  and  El  normally  in- 
ward (that  is,  —  .Zfj).  Hence 

0  =  E2  -  El 
Therefore  El^E2  =  \E 

The  charge  upon  the  element  of  surface  dS  is  crdS  =  DdS ; 
and,  since  the  intensity  at  the  charged  element  due  to  crdS  is 
zero  (being  directed  symmetrically  toward  the  outside  and  in- 
inside),  and  since  the  intensity  due  to  the  other  charges  is  ^E,  the 
mechanical  force  per  unit  area  upon  the  charged  surface  between 
P  and  \S  \*IC-«C.  (3.) 


GENERAL  ELECTROSTATIC  THEORY.         35 

Electric  Tension,  Method  III.  The  same  result  may  be  ob- 
tained by  still  another  method.  As  we  have  seen,  the  electric 
charge  is  not  strictly  a  surface  distribution,  but  is  confined  to 
a  very  thin  surface  layer.  At  the  outer  surface  of  the  layer 


Charged 


as 

i 


Fig.  9. 

the  intensity  and  displacement,  which  are  normal  to  the  surface 
throughout  the  layer,  have  their  full  surface  values  E  and  D  ;  at 
the  inner  boundary  of  the  layer  they  are  zero.  If  E  and  D  de- 
note also  the  intensity  and  displacement  at  a  distance  x  from  the 
inner  boundary  of  the  surface  layer,  of  thickness  L  (Fig.  9),  the 
charge  within  the  small  volume  of  thickness  dx  and  cross-section 

dq  •=  dx  dS  divZ>  =  dx  dScdEj  dx 

E  being  a  function  of  x  only.  The  outward  force  upon  the  por- 
tion of  the  conductor  within  this  charged  volume  is 

dF'  =  Edq  =  dS  cEdE\dxdx 

and  the  total  force  upon  that  part  of  the  surface  layer  whose 
cross-section  is  dS  is 


IdF9  =  dS  f  cEdEldxdx^dSc  f 
•J  i7o  *A) 


dF  =   IdF9  =  dS       cEdEdxdx^dSc       EdE=dS\cE* 

whence  the  force  per  unit  area  upon  the  charged  surface,  or  the 
tension  in  the  dielectric  at  the  surface,  is 

r=  dFjds  =  \cE*  =  \ED  (3  1) 

41.  Electric  Pressure.     Equilibrium  of  a  Dielectric  Supporting 
Electric  Displacement.      In  a  dielectric   supporting  an    electric 


30  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

field  there  is  at  every  point,  in  addition  to  the  tension  }>ED  in 
the  direction  of  the  intensity,  a  pressure  normal  in  every  direc- 
tion to  this  intensity  and  equal  to 


•  T=  u  (32) 

To  establish  this  proposition,  consider  the  radial  field  from  a 
charge  upon  a  very  small  body  at  P,  Fig.  10,  and  an  elementary 
(conical)  tube  of  displacement  T  cutting  two  (spherical)  equipo- 


S1 


Fig,  10. 

tentials  S^  and  S^  a  distance  dL  apart,  and  enclosing  areas  dSl 
and  dS2  of  these  surfaces.  Let  Ev  Dl  and  £2,  D2  be  the  inten- 
sity and  displacement  at  Sl  and  S2,  respectively.  The  portion  Z 
of  the  dielectric  enclosed  by  the  sides  of  the  tube  and  Sl  and  S2 
is  in  equilibrium  under  the  action  of  the  stresses  of  the  field. 
The  force  on  Z  arising  from  the  tensions  is,  by  (31), 


measured  toward  P,  and  must  be  balanced  by  an  equal  force 
directed  from  P.  We  shall  assume  that  this  equilibrating  force 
arises  from  a  pressure  p  normal  everywhere  to  the  surface  of  the 
tube,  and  shall  proceed  to  find  its  value.  If  p^  and  pz  are  the 
values  of  p  at  dS^  and  dS^  respectively,  its  average  value  over 
the  surface  of  the  small  volume  Z  is  approximately 

KA+A) 


GENERAL   ELECTROSTATIC    THEORY.  37 

From  the  figure  it  is  clear  that  the  resultant  force  due  to  /  is 
outward  along  the  axis  of  the  tube  and  equal,  approximately,  to 


if  B  is  the  lateral  area  of  the  surface  of  Z.     Since 

sin  6  =  (dS2  -  dS 
very  approximately, 

sin  6  B  =  \  (A 


Z  is  evidently,  by  symmetry,  in  equilibrium  laterally,  so  that 
it  will  be  in  complete  equilibrium  if 

K  A  +  A)  (A  -  A)  =  i^'A  - 

Since 


the  last  equation  may  be  written 

K  A  +  A)  (A  -  A)  =  i  ^A(  A  -  A) 


which  becomes,  when  the  tube  T  is  made  indefinitely  narrow, 
and  the  surfaces  Sl  and  S2  are  brought  indefinitely  close  together, 


which  is  identical  with  (32). 

Since  the  field  within  the  element  of  volume  is  uniform  when 
the  element  is  made  indefinitely  small,  and  since  this  is  true  of 
any  electric  field,  the  result  just  obtained  for  a  radial  field  holds 
universally. 

A  method  of  proving  (32)  by  direct  experiment  is  described 
in  §  3>  VII. 

42.  Electric  Conduction  and  Induction.  The  tension  in  the 
direction  of  the  intensity  at  every  point  of  a  dielectric  supporting 
an  electric  field  and  the  pressure  perpendicular  to  this  direction 
throw  much  light  on  the  disappearance  or  transfer  of  electric 
charges  by  conduction  and  their  development  by  induction. 


38     ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

Conduction,  First  it  will  be  shown  that  this  stress  system  will 
account  for  the  result  of  §  26.  For  if  the  surface  of  a  conductor 
were  not  an  equipotential,  i.  e.t  if  the  tubes  of  induction  did  not 
meet  the  surface  normally,  there  would  be  in  the  dielectric  at  the 
interface  a  component  of  the  intensity  parallel  to  the  surface,  and 
therefore  a  component  of  the  tension  parallel  to  the  surface 
and  a  component  of  the  pressure  perpendicular  to  the  surface. 
Since  the  charges,  or  the  ends  of  the  tubes,  can  move  freely  along 
a  conductor,  the  tubes  would  therefore  contract,  their  ends,  or 
the  charges,  slipping  along  the  conductor ;  and  since  within  the 
substance  of  the  conductor  the  intensity,  and  therefore  the  pressure 
perpendicular  to  the  surface,  is  zero,  the  component  perpendicular 
to  the  interface  of  the  pressure  in  the  dielectric  would  be  un- 
balanced by  any  pressure  from  within,  so  that  the  tubes  would 
be  continually  pushed  toward  and  into  the  conductor  (there  to 
break  up).  Since  these  processes  are  inconsistent  with  the  nature 
of  a  static  field,  there  can  be  no  component  of  the  intensity 
parallel  to  the  surface  of  the  conductor. 

Consider  two  electrified  conductors  A  and  B  with  positive 
and  negative  charges  respectively,  A  being  the  only  positively 
charged  body  in  the  field  ;  and  suppose  q,  the  charge  of  A,  nu- 
merically greater  than  q1 ',  the  charge  of  B.  Of  the  q  unit  tubes 
emanating  from  A,  q'  terminate  upon  B,  and  q  —  q'  upon  other 
bodies  at  a  distance.  If  A  and  B  are  connected  by  a  wire  C,  in 
which  permanent  electric  stress  cannot  exist,  the  tensions  along 
the  tubes  and  pressures  at  right  angles  to  them  will  cause  the  q' 
tubes  connecting  A  and  B  to  be  pushed,  contracting  as  they  go, 
into  the  regions  of  no  permanent  stress,  the  conductors  A,  B, 
and  C,  until  the  positive  and  negative  ends  of  each  tube  meet 
and  the  tube  disappears. 

The  remaining  q  —  q'  tubes  emanating  from  A  will  be  redis- 
tributed by  the  system  of  tensions  and  pressures  until  there  is 
again  equilibrium,  when  the  remaining  q  —  q'  tubes  will  emanate 
normally  from  A,  B,  and  C. 


GENERAL  ELECTROSTATIC  THEORY.         39 

During  the  process  of  conduction  the  field  is  not  in  equilib- 
rium, nor  is  it  zero  within  the  conductors,  and  the  tubes  are  not 
normal  to  the  surfaces  of  the  conductors,  but  are  inclined  from 
the  normal  at  each  end  in  the  direction  of  motion  of  that  end. 
If  the  conductivity  were  perfect,  the  tubes  would  always  end 
normally  at  the  conducting  surfaces  and  would  never  disappear 
in  the  conductors  (Chapter  VIII.,  §  9). 

The  phenomena  here  described  are  only  a  part  of  the  phe- 
nomena occurring  during  conduction,  and  a  more  complete  dis- 
cussion will  be  given  later  (Chapters  VIII.,  XIL). 

Induction.  Into  an  electric  field,  as  that  bounded  by  a  concen- 
trated charge  A  and  the  walls  of  the  room,  let  a  conductor  B  be 
introduced.  The  state  of  strain  previously  existing  in  the  space 
now  occupied  by  B  is  annulled  by  its  introduction,  the  tubes 
formerly  crossing  this  space  being  cut  in  two  by  the  conductor,  and 
those  sufficiently  near  being  pushed  against  its  surface  and  there 
also  cut  in  two,  until  all  the  tubes  so  severed  touch  B  normally 
and  there  is  again  equilibrium.  For  every  tube  terminating  upon 
B  there  is  therefore  a  tube  of  equal  strength  emanating  from  it. 
That  is,  the  positive  and  negative  charges  developed  by  induction 
are  equal. 

All  the  tubes  severed  by  B  may  be  regarded  as  still  belonging 
to  A  ;  they  are  simply  rendered  discontinuous  at  the  surface  of 
B,  where  the  induced  charges  therefore  reside. 

If  B  is  connected  to  the  walls  by  a  wire  C,  the  tubes  stretching 
from  B  to  the  walls  will  disappear  by  the  process  of  conduction, 
described  above,  leaving  B  charged  oppositely  to  A.  The  tubes 
between  B  and  the  walls  having  disappeared,  more  of  the  tubes 
from  A  will  crowd  into  their  places  until  there  is  again  equilib- 
rium, part  to  end  on  B,  part  on  C,  and  part  on  the  walls.  The 
charge  on  B  of  the  opposite  kind  to  that  of  A  is  thus  increased 
by  earthing  B. 

Fig.  1 1  (from  Nichols  and  Franklin's  Elements  of  Physics, 
Vol.  II.,  §§  165-6)  illustrates  the  process  of  introducing  a  small 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


charged  conductor  B,  insulated,  into  a  nearly  closed  hollow  con 
ductor  Ay  §4,  putting  on  the  conducting  lid,  moving  B  about 
inside,  and  finally  bringing  B  into  contact  with  A's  inner  surface. 
The  distribution  of  the  tubes  here,  as  well  as  in  the  preceding 
cases,  can  be  roughly  predicted  from  the  considerations  that  all 
the  tubes  meet  both  conductors  normally  and  that  the  voltage 


Fig.  11. 

along  every  line  from  one  conductor  to  the  other  is  the  same. 
Since  the  voltage  along  a  line  of  intensity  is  equal  to  the  average 
value  of  the  intensity  along  the  line  x  its  length,  the  intensity 
and  the  induction  must  be  greater,  or  the  tubes  more  concen- 
trated, the  shorter  the  distances  through  which  they  stretch. 

If,  instead  of  a  conductor,  an  insulator  of  permittivity  different 
from  that  of  the  dielectric  in  the  region  is  introduced  into  the 


GENERAL  ELECTROSTATIC  THEORY.         41 

field,  phenomena  similar  in  some  respects  are  observed.     This 
subject  will  receive  consideration  in  Chapter  IV. 

43.  Electric  Tension  and  Pressure  and  Forcives  in  the  Electric 
Field.     The  force  upon  a  conductor  in  or  bounding  a  static  field 
is  due  wholly  to  the  tension  in  the  dielectric,  the  force  per  unit 
area  at  any  point  of  the  surface  being  T  normal  to  the  surface. 
Since  E  and  D  are  perpendicular  to  a  conductor's  surface,  no 
component  of  the  force  upon  a  conductor  can  be  due  to  the  elec- 
tric pressure,  which  is  tangential  to  the  surface  and  balanced  in 
every  direction.      Inasmuch,  however,  as  the  distribution  of  the 
tubes,  and  thus  the  distribution  of  T,  over  the  surface  of  a  con- 
ductor is  determined  by  both  tensions  and  pressures,  the  latter 
contribute  to  the  force  indirectly. 

The  force  upon  a  dielectric  is  in  general  due  to  both  tensions 
and  pressures.  See  Chapter  IV.,  §  9. 

In  Chapers  II.,  III.,  IV.,  VI.  and  VII.  many  examples  will  be 
found. 

44.  The  Equilibrium   of   a  Given  Field  is  not  Altered  if  its 
Direction  is  Reversed  at  Every  Point.     For  the  tension  parallel 
to  the  intensity  and  the  pressure  perpendicular  to  the  intensity, 
at  any  point  of  the  field  are  proportional  to  its  square,  and  are 
therefore  not  altered  by  the  reversal  of  direction.     The  signs  of 
all  charges  are  of  course  reversed  with  the  reversal  of  the  in- 
tensity. 

45.  When  the  Algebraic  Charge  upon  Each  Conductor  of  an 
Isolated  System  is  Zero,  there  is  no  Charge  and  no  Field.     If  the 
total  algebraic  charge  of  each  of  a  system  of  conductors  is  zero, 
and  if  there  are  no  other  electrified  bodies,  or  electrets  (Chapter 
VI.),  or  a  changing  magnetic  flux  (Chapter  XIII.)  anywhere, 
then  there  is  no  electric  charge  or  displacement  anywhere.      For 
if  the  electric  field  were  not  zero,  tubes  of  displacement  would 
emanate  from  each  conductor  and  terminate  upon  conductors  of 
lower  potential,  since  each  conductor  would  have  both  positive 
and  negative  charges.     And  since  the  algebraic  charge  of  each 


42  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

conductor  is  zero,  as  many  tubes  as  emanated  from  the  conductor 
at  highest  potential  would  terminate  upon  its  surface,  while  there 
would  be  no  body  at  higher  potential  at  which  such  tubes  could 
originate  ;  and  as  many  tubes  as  terminated  upon  the  conductor 
at  lowest  potential  would  emanate  from  its  surface,  while  there 
would  be  no  body  at  lower  potential  for  these  tubes  to  terminate 
upon.  Thus  the  supposed  case  is  impossible,  and  there  is  no 
charge  or  intensity  anywhere. 

46.  A  Single  Electric  Field   Corresponding  to  Given  Charges. 
If  an  electric  field  Al  bounded  by  a  system  of  fixed  conductors 
with  given  total  charges,  or  by  a  system  of  insulators  with  both 
charges  and  their  distribution  given,  or  by  both,  is  in  equilibrium, 
this  is  the  only  field  satisfying  the  given  conditions  which  is  in 
equilibrium.      For  suppose  that  A2  is  a  second  field  satisfying  the 
conditions  :  It  will  be  shown  that  y42  =  Ar     For  if  A2  with  its 
sign  reversed  is  superposed  upon  Av  the  resultant  field  will  be 
in  equilibrium,  and  the  charge  at  each  point  of  every  insulator 
and  the  total  charge  of  each  conductor  (the  last  statement  in 
§  32  not  being  assumed  as  known)  will  be  zero.      Hence  by  the 
last  article  there  is  no  electric  charge  or  displacement  anywhere. 
Thus  at  every  point  Dl  -f  (—  Z>2)  =  o,  or  A2  =  Ar 

A  Single  Field  Corresponding  to  Given  Potentials.  If  an  elec- 
tric field  Av  bounded  by  a  given  system  of  conductors,  the  po- 
tential of  each  being  given,  or  the  voltage  between  each  and 
all  the  rest,  is  in  equilibrium,  this  is  the  only  equilibrium  field 
satisfying  the  given  conditions.  For  let  A2,  another  field,  sup- 
posedly, satisfying  the  conditions  be  superposed  upon  Al  with  its 
sign  reversed.  In  the  resultant  field  the  potential  of  each  con- 
ductor, or  the  voltage  between  each  and  all  the  rest,  is  zero,  by 
the  last  article.  Hence  the  intensity  is  zero  everywhere,  and 
A,  =  A, 

* 

47.  Equipotential  Replaced  by  Infinitely  Thin   Conductor  of 
Same  Shape.      In  any  electric    field    any   equipotential    surface 


GENERAL  ELECTROSTATIC  THEORY.         43 

vS  (always  closed  or  else  an  infinite  plane)  can  be  replaced  by  an 
infinitely  thin  conducting  sheet  Sr  without  disturbing  the  electric 
field  on  either  side.  For  all  the  tubes  which  before  the  substi- 
tution crossed  .S  normally,  after  the  substitution  terminate  nor- 
mally on  one  side  of  Sf  and  emanate  normally  from  the  other 
side  (induced  charges  being  developed).  But  since  Sf  is  coinci- 
dent with  S,  this  necessitates  no  change  in  the  direction  or  posi- 
tion of  any  tube,  and  the  substitution  therefore  leaves  the  field 
in  undisturbed  equilibrium,  the  tensions  and  pressures  remaining 
precisely  the  same  as  before  the  substitution.  This  result  is 
also  a  corollary  of  the  next  article. 

Definition  of  Electric  Images.  The  two  charges,  or  systems 
of  charges,  in  the  regions  on  opposite  sides  of  S'  (but  not  in- 
cluding the  charges  upon  Sf)  are  called  the  electric  images  of  one 
another  in  the  surface  S' . 

Since  the  conducting  sheet  S'  renders  the  fields  on  its  opposite 
sides  absolutely  independent  of  one  another,  either  field  may  be 
destroyed  or  modified  in  any  manner  without  affecting  the  other. 
If  we  are  concerned  with  only  one  of  these  fields,  the  substance 
of  the  conductor  whose  surface  S'  coincides  with  5  may  be  ex- 
tended in  any  manner  into  the  region  (previously)  occupied  by 
the  other  field. 

48.  Additional  Propositions  Fundamental  to  the  Method  of  Elec- 
tric Images.  If  the  potential  of  a  surface  vS  and  the  charges  on 
one  side  of  5  (closed  or  an  infinite  plane,  being  an  equipotential) 
are  given,  the  electric  field  on  this  side  is  fixed  independently  of 
the  way  in  which  the  surface  is  kept  at  the  given  potential.  For 
suppose  that  two  fields  A^  and  A2  satisfy  the  conditions.  If  A2 
with  its  direction  everywhere  reversed  is  superposed  upon  Av  S 
will  be  at  zero  potential,  and  there  will  be  no  electric  charge  on 
the  side  of  S  considered.  Hence  there  is  no  field  on  this  side, 
and  A2  =  AY 

Similarly,  if  the  position  of  an  equipotential  surface  ,S  is  given, 
together  with  the  flux  across  it  and  the  charges  on  one  side  of  it, 


44  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

the  field  on  this  side  is  fixed  and  independent  of  the  way  in  which 
5  is  kept  equipotential  and  of  the  way  in  which  the  flux  across 
it  is  kept  of  the  given  magnitude.  For  suppose  that  two  fields 
Al  and  A2  satisfy  the  conditions.  If  —  A2  is  superposed  upon 
Aiy  the  surface  5  will  still  be  an  equipotential,  the  flux  across  it 
will  be  algebraically  zero,  and  there  will  be  no  charge  on  the 
side  of  vS  considered.  Hence  not  only  the  total  flux  across  vS  is 
zero,  but  the  flux  across  every  part  of  S,  and  there  is  no  field  on 
the  side  of  .S  considered.  Thus,  as  before,  A2  =  Ar 

In  the  above  two  cases,  if  the  given  charges  are  upon  con- 
ductors, only  the  total  charges  need  be  given  ;  but  if  the 
charges  are  upon  insulators,  the  distribution  as  well  as  the  mag- 
nitude of  each  charge  must  be  given. 

49.  The  Electrostatic  Energy  of  an  Electric  Field  Surrounding 
any  Number  of  Conductors.  We  shall  now  proceed  to  find  the 
energy  of  an  electric  field  containing  any  number  of  conductors, 
Av  Ay  -  •  -,  An,  with  any  charges  qlf  qv  •  -  •,  qn,  and  at  any  po- 
tentials Vv  Vv  •  •  •,  Vn\  Vl  and  Vn  being  the  highest  and  lowest 
potentials,  respectively.  For  the  sake  of  keeping  the  field 
within  finite  limits  and  eliminating  the  field  of  the  earth,  let  all 
the  conductors  Av  A2,  etc.,  be  enclosed  within  a  hollow  closed 
conductor  Ao,  such  as  the  walls  of  a  room,  at  potential  Vo.  This 
limitation  will  be  removed  later  for  ideal  cases.  Some  of  the 
conductors  will,  in  general,  be  at  higher,  and  some  at  lower  po- 
tentials than  Vo.  Let  the  field  be  divided  into  regions  of  higher 
and  lower  potential  than  Vo  (by  equipotential  surfaces  of  poten- 
tial  F). 

Consider  first  the  conductors  Ad,  Ae,  •  •  •,  in  a  region  above  Vo 
in  potential.  On  Ad,  the  conductor  at  highest  potential  in  the 
region,  no  tubes  terminate,  and  from  it  qd  tubes  emanate,  some 
terminating  on  Ao  and  others  cutting  the  part  of  the  equipotential 
V  separating  the  region  under  consideration  from  the  neighbor- 
ing region  at  lower  potential.  Some  of  the  qd  tubes  are  discon- 
tinuous at  the  surfaces  of  the  conductors  A^  Ajy  etc.,  in 'the  region 


GENERAL  ELECTROSTATIC  THEORY.         45 

(the  discontinuities  corresponding  to  induced  charges),  but  all 
finally  reach  the  boundary,  at  potential  F,  of  the  region.  The 
voltage  along  every  tube  from  Ad  to  the  boundary  of  the  region 
is  therefore  Vd—Vo.  The  energy  contributed  to  the  region  by  the 
tubes  emanating  from  Ad  is  thus  \qJ^Vd  —  F).  Since  Ae  pos- 
sesses a  charge  qe,  qe  unit  tubes  emanate  from  Ae  (in-  addition  to 
the  tubes  from  Ad,  which  both  terminate  upon  and  emanate  from 
Ae,  and  have  already  been  considered),  and  all  finally  reach  the 
boundary  of  the  region  at  potential  F.  The  energy  contributed 
to  the  region  by  the  qe  unit  tubes  from  Ae  is  thus  ±qe  (Ve—  V^)\ 
and  so  on  for  the  other  conductors  in  the  region. 

Consider  now  a  region  of  lower  potential  than  F  containing 
conductors  •  •  -,  A.,  Ak,  Ap  Al  being  the  conductor  at  lowest  poten- 
tial in  the  region.  No  tubes  emanate  from  Aiy  and  the  ql  unit 
tubes  terminating  upon  it  all  come  from,  some  through,  the 
boundary  of  the  region  at  potential  F,  though  some  are  discon- 
tinuous (induced  charges)  at  the  surfaces  of  Ak  and  the  other  con- 
ductors, at  higher  potentials  than  Vv  in  the  region.  The  voltage 
of  each  of  these  tubes  from  the  boundary  to  Al  is  thus  F  —  Vlt 
and  since  the  total  strength  of  all  the  tubes  is  —  q  (q  being  nega- 
tive since  Vl  is  less  than  Fo),  the  energy  contributed  to  the  region 
by  the  tubes  of  Al  is  \ql  (Vl—  F).  The  energy  contributed  by 
the  qk  tubes  belonging  to  A  k  is  likewise  \qk(Vk—  F)  ;  and  so 
on  for  the  other  conductors  of  this  region,  and  for  other  regions. 

Summing  up  these  expressions  for  the  whole  electric  field 
within  Ao,  we  have  for  the  total  energy  within  Ao 


fO          (33) 

If  Ao  is  connected  to  the  earth,  and  if  we  define  the  potential 
of  the  earth  as  zero  potential,  (33)  becomes 

W=\T.qV  (33«) 

If  we  suppose  Ao  removed  to  infinity,  if  we  suppose  the  n  con- 
ductors to  be  the  only  electrified  bodies  in  space,  and  if  we  define 


4^  ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

the  potential  of  Ao,  in  a  region  infinitely  remote  from  all  electrified 
bodies,  as  zero  potential,  (33*2)  will  give  the  total  electrical  energy 
in  space  surrounding  this  ideal  system. 

50-56.  A  System  of  Conductors  Av  A2,  •  •  •,  An  Surrounded  by  a 
Closed  Conductor  Ao.  The  voltage  from  any  conductor  of  the 
system,  such  as  Ak,  to  Ao  will  be  denoted  by  V  with  the  proper  sub- 
script, as  Vv  and  its  charge  by  q  with  the  same  subscript,  as  qk. 

50.  Voltages  in  Terms  of  Charges  and  Charges  in  Terms  of  Volt- 
ages. The  voltage  Vk  from  any  one  Ak  of  the  system  of  con- 
ductors to  Ao  is  a  linear  function  of  the  charges  of  all  the  con- 
ductors. For,  by  §§34,  35,  Vk  is  proportional  to  the  charge  of 
any  one  conductor  of  the  system  Av  A2,  •  ••,  An  when  all  the 
rest  are  insulated  without  charge  (except  induced  charges),  and 
therefore,  by  §  33,  when  all  the  conductors  are  charged  the  ex- 
pression for  Vk  consists  of  a  series  of  terms  each  proportional  to 
the  charge  of  one  conductor.  Hence 

\q  (i) 

in--*  n  \     / 

(34) 


The  coefficients  of  the  charges,  viz.,  /u,  /12,  etc.,  are  called 
voltage  coefficients,  or  coefficients  of  potential.  Each  has  two  sub- 
scripts, the  first  identical  with  that  of  the  conductor  in  the  ex- 
pression for  the  voltage  from  which  to  Ao  it  occurs,  the  second 
identical  with  that  of  the  conductor  to  whose  charge  the  coef- 
ficient belongs,  and  denotes  the  ratio  of  the  voltage  to  Ao  from 
the  conductor  with  the  first  subscript  to  the  charge  of  the  con- 
ductor with  the  second  subscript  when  all  the  conductors  except 
that  with  the  second  subscript  are  insulated  without  (algebraic) 
charge.  Thus  for  example, 


when  qv  qv  -  •  -,  qn_^  are  zero  in  (34)  (2). 


GENERAL  ELECTROSTATIC  THEORY.         47 

By  solving  the  n  equations  (34),  each  of  the  charges  may  be 
expressed  as  a  linear  function  of  all  the  voltages.     Thus 


(0 

(2) 


(35) 


the  s's  being  functions  of  the  />'s. 

Each  s  has  two  subscripts  and  denotes  the  ratio  of  the  charge 
upon  the  conductor  with  the  first  subscript  to  the  voltage  to  Ao 
from  the  conductor  with  the  second  subscript  when  the  voltage 
to  A0  from  each  of  all  the  conductors  except  that  with  the  second 
subscript  is  zero  (all  the  conductors  except  that  with  the  second 
subscript  connected  metallically  with  Ao). 

Those  coefficients  with  the  two  subscripts  equal  are  called 
coefficients  of  capacity,  while  those  with  the  subscripts  unequal 
are  called  coefficients  of  induction. 

The  /'s  and  s's  are  functions  only  of  the  configuration  of  the 
conductors  and  of  the  dielectric  constant  of  the  medium  in  which 
the  system  is  placed. 

51.  The  Coefficients.  Any  two  coefficients  of  potential  with 
the  same  subscripts  in  different  order  are  identical.  That  is 

Pu=Piu  (36) 

To  prove  this  relation,  let  all  the  conductors  of  the  system 
Al  •  •  •  An  except  Ah  and  Al  remain  insulated  and  uncharged  (alge- 
braically). The  energy  of  the  dielectric  within  Ao  when  Ah  and 
Al  have  the  charges  qh  and  ql  will  be  independent  of  the  manner 
in  which  the  field  is  established.  The  energy  obtained  by 
charging  Ah  first  and  then  Al  may  therefore  be  equated  to  that 
obtained  by  charging  Al  first  and  then  Ah.  Denoting  the  energy 
in  the  first  case  by  Wl  and  that  in  the  second  case  by  W2,  we 
have 


48  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

W,  =    f^  Vhdgh  fa  =  o]  +    P  Vfr,  fa  =  ?  J 

t'O  */0 

=    f  A^A+    f   (A*ft 

t/O  «^0 


+ 


X?J  /»9/ 

Pu<2id<li+  I    (Atfi 
«^o 


from  which  (36)  immediately  follows. 

If  moreover  the  expressions  in  /'s  for  any  two  coefficients  of 
capacity  with  the  same  subscripts  in  opposite  order,  e.  g.,  shl  and 
slh,  are  examined,  it  will  be  found  that  they  differ  only  in  having 
such  coefficients  of  potential  as  phl  and  plh  interchanged.  But 

PM  =  Pw  nence 

**  =  *»  (37) 

With  the  aid  of  (34)  and  (35)  these  results  may  be  interpreted 
as  follows  : 

(36)  The  voltage  to  Ao  from  any  conductor  Ah  when  Al  has  a 
given   charge   q  and   all   the  other   conductors   of  the   system 
Al  -  -  •  An,  including  Ah,  are  insulated  without  algebraic  charge, 
is  equal  to  the  voltage  from  At  to  Ao  when  Ah  has  the  same 
charge  q  and  all  the  other  conductors,  including  Av  are  without 
charge. 

(37)  The  charge   upon  any  conductor  Ah  when  the  voltage 
from  any  other  conductor  Al  to  Ao  is  F,  and  the  voltage  to   Ao 
from  each  of  all  the  other  conductors,  including  Ah,  is  zero  (all 
the  conductors  except  Al  connected  metallically  to  Ao),  is  equal 
to  the  charge  upon  Al  when  the  voltage  from  Ah  to  Ao  is  Vt  and 
the  voltage  to  Ao  from  each  of  all  the  other  conductors,  includ- 
ing Alf  is  zero  (all  the  conductors  except  Ah  connected  to  Ao). 

The  coefficients  of  voltage  are  all  positive.  For  if  one  con- 
ductor has  a  positive  charge,  all  the  rest  being  uncharged  (alge- 
braically), lines  of  intensity  emanate  from  it  and  pass  to  Ao,  some 


GENERAL  ELECTROSTATIC  THEORY.         49 

of  them  crossing  (discontinuously)  the  other  conductors  as  they 
go.  Thus  lines  pass  from  each  conductor  to  Ao.  Hence  the 
voltage  from  any  conductor  to  Ao  is  greater  than  zero,  or  posi- 
tive, and  the  coefficients  of  voltage  are  therefore  positive 

(  (34)  ff.). 

The  coefficients  of  capacity  are  all  positive,  and  the  coefficients 
of  induction  are  all  negative.  For  if  any  conductor  is  so  charged 
that  the  voltage  from  it  to  Ao  is  positive,  while  the  voltage  to  Ao 
from  each  of  all  the  rest  is  zero  (the  conductors  connected  to  Ao), 
lines  of  intensity  pass  from  this  conductor  to  all  the  others. 
Hence  its  charge  is  positive,  and  the  charges  of  all  the  rest  are 
negative.  Hence  the  coefficients  of  capacity  are  all  positive,  and 
the  coefficients  of  induction  are  all  negative  (  (35)  ff.). 

The  coefficient  of  capacity  of  any  conductor  Ah  is  numerically 
equal  to,  or  greater  than,  the  sum  of  all  the  coefficients  of  in- 
duction (with  h  as  first  subscript)  between  Ah  and  the  other  con- 
ductors. For  when  Ah  is  insulated  and  charged,  and  the  voltage 
to  Ao  from  each  of  all  the  other  conductors  (connected  to  A^  is 
zero,  the  number  of  unit  tubes  ending  on  the  other  conductors 
cannot  be  greater  than  the  number  emanating  from  Ah  (since  all 
emanate  from  Ah),  and  can  equal  this  number  only  when  Ah  is 
completely  surrounded  by  one  or  more  of  the  other  conductors 
(of  the  system  Av  Av  •  •  •,  An]  as  it  is  surrounded  by  Ao  (or 
when  Ao  is  removed  to  infinity  and  the  other  conductors  are  the 
only  two  charged  bodies  in  space). 

When  the  dielectric  is  homogeneous  and  isotropic  throughout, 
the  coefficients  of  induction  and  capacity  are  proportional,  and 
the  voltage  coefficients  inversely  proportional,  to  its  permittivity. 

Every  voltage  coefficient  with  its  subscripts  equal,  as  phh,  is 
diminished,  and  every  coefficient  of  capacity,  as  shh,  is  increased, 
by  the  introduction  of  another  conductor  into  the  field.  For  Vh, 
the  voltage  from  Ah  to  Ao  is  equal  to  § ELdL  along  any  path 
from  A,  to  A  ;  and  when  another  conductor  is  introduced, 

li  o  ' 

whether  it  is  insulated  or  connected  to  Ao,  tubes  of  displacement 
are  pushed  toward  and  into  this  conductor,  making  the  field  less 


50  ELEMENTS  OF    ELECTROMAGNETIC    THEORY. 

intense,  and  j  ELdL  less,  along  some  paths  not  passing  through 
the  new  conductor  (and  therefore  along  all  paths).  Hence, 
while  qh  remains  unaltered,  Vh  is  diminished  by  the  introduction 
of  the  new  conductor.  When  Ah  is  insulated  with  charge  qht  and 
all  the  other  conductors  are  insulated  without  algebraic  charge, 
phh  =  Vh  jqhJ  and  is  therefore  diminished  by  the  introduction  of 
the  new  conductor.  When  Ah  is  insulated  with  charge  gh  and 
the  voltage  to  Ao  from  each  of  all  the  other  conductors  (con- 
nected to  Ao]  is  zero,  s^  =  qh  /  Vh  and  is  therefore  increased  by 
the  introduction  of  the  new  conductor. 

It  is  easy  to  see  that  the  effect  in  question  is  greater  in  each 
case  the  nearer  the  new  conductor  is  brought  to  the  conductor 
whose  coefficients  are  under  consideration,  and,  in  general,  the 
greater  its  volume  (space  included  within  its  exterior  surface,  if 
hollow). 

Similar  and  opposite  effects,  respectively,  are  produced  by  in- 
troducing into  a  part  of  the  field  a  dielectric  of  greater  or  less 
permittivity  than  that  of  the  rest  of  the  dielectric  within  Ao,  as 
will  be  apparent  after  reading  Chapter  IV. 

52.    Additional   Expressions    for   the    Electric    Energy.      The 

energy  of  the  electric  field  within  Ao  surrounding  the  system  of 
conductors  Av  A2,  •  •  •  ,  An  (§49)  can  also  be  expressed  as  a 
quadratic  function  of  all  the  charges  gv  •  •  •  ,  qn,  or  of  all  the 
voltages  Vv  -  -  •  ,  Vn.  For  by  (30)  and  (35) 


.U  ,    g) 

+  ^33+-"+^2K  +  -" 
and  by  (30)  and  (34) 


Wv  denoting  the  energy  expressed  in  terms  of  the  voltages,  and 
Wq  the  energy  expressed  in  terms  of  the  charges. 
By  partial  differentiation  we  find  that 


GENERAL   ELECTROSTATIC    THEORY.  51 

WJ<tri-sliri  +  sur,+  ...  +  s^rn 
=  qv  dWJdVz=  qv  etc,  and 

dWJdq,  =  pnq,  +  plzq2  +  .  .  .  +  Plnqn  =  Vv  dWJdqz  =  Vv  etc.  (4  1  ) 


53.  2Vq'  =  2V'q.  Let  qv  Vv  qv  F2,  .  .  .  ,  ?n,  F,  and  «?/,  F/, 
£/>  ^'»  •"»£/»  ^V  denote  the  charges  and  voltages  to  ^4o  for 
two  static  fields  surrounding  the  system  of  conductors  Alt  •  •  •  , 
A  within  A  .  Then 

EF/  =  2FV  (42) 

For  2  F/  =  J^/  +  V#i  -f  •  -  •  +  ^/.     Whence,  by  (34), 


and 

by  adding  up  in  vertical  columns  the  corresponding  terms. 

54.  Change  in  Energy  when  Charges  and  Voltages  are  Altered. 
If  the  charges  and  voltages  to  Ao  of  the  given  system  of  con- 
ductors are  changed  from  one  set  of  values  qlt  Vv  q2,  Vv  etc.,  to 
another  set  ^/,  F/,  #/,  F2r,  etc.,  the  increase  in  the  energy  of  the 


field  is  W  -  W  =  i  (2/  F'  -  ^q  F)  (43) 

This  equation  may  be  put  in  two  other  forms,  sometimes  con- 
venient, by  (42).     Thus 


F)  (44) 

V  +^qV-  2?'  F-  2?  F) 

F).  (45) 


55.  Electric  Energy,  Mechanical  Energy,  and  Change  of  Con- 
figuration.   From  §53  it  follows  that  if  the  system  of  conductors 


52  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

Alt  A2,  •  ",  An  suffers  a  certain  change  of  configuration,  the 
charges  qv  qv  etc.,  remaining  constant,  and  the  voltages  to  Ao 
therefore  changing  from  Vv  Vv  etc.,  to  F/,  F2',  etc.,  the  energy, 
Wt  lost  by  the  electric  field  minus  the  energy,  W  ',  gained  by  the 
field  when  the  same  change  of  configuration  occurs  with  voltages 
Vlt  Vv  etc.,  constant,  and  charges  therefore  changing  from  qv  qv 
etc.,  to  qft  q£  ,  etc.,  is  equal  to 

W-  W  =  $2q(  V-  V)  -  £2  V(q'  -  q) 

=  i2(/-^)(F'-F)  (46) 

For  in  the  first  case,  after  the  change  of  configuration,  the 
charges  qv  qv  etc.,  correspond  to  the  voltages  F/,  F/,  etc.;  and 
in  the  second  case,  after  the  same  change  of  configuration,  the 
charges  ^/,  q£,  etc.,  correspond  to  the  voltages  Vv  Vv  etc. 
Hence  by  (42) 


Substituting  2^'  V  for  ^q  V  in  the  first  term  of  the  central 
member  of  (46),  we  obtain  £(2?'  V  -  2?F'  +  ^qV  -  2</'  F), 
which,  on  being  factored,  becomes  identical  with  the  last  member 
of  (46). 

If  the  energy  dissipated  in  heat  during  the  change  of  con-» 
figuration  (owing  to  electric  resistance  (VIII.))  and  that  radiated 
away  (both  of  which  are,  or  may  be  made,  exceedingly  small) 
are  neglected,  the  system  gains  in  the  first  case  an  amount  of 
mechanical  energy  equal  to  W,  the  loss  of  electric  energy.  If 
now,  after  the  change  of  configuration,  the  voltages  to  Ao  are 
brought  back  to  their  initial  values  by  means  of  batteries  (or  other 
agents  possessing  intrinsic  e.m.f.s  (VIII.)),  the  state  of  the  system 
will  be  the  same  as  after  the  change  of  configuration  in  the  second 
case  above,  and  the  electrical  energy  will  surpass  the  initial 
energy  by  the  amount  W  .  The  energy  W"  supplied  to  the 
system  by  the  batteries  is  equal  to  the  sum  of  the  increases  in 
mechanical  and  electrical  energy,  or 

W"  = 


GENERAL   ELECTROSTATIC   THEORY.  53 

But  W'=W-\*df 

hence  W"  =  2W  -  £2  (qf  -  q)  (V  -  V)  (47) 

If  the  change  of  configuration  is  infinitesimal,  Wt  Wf,  and  W" 
are  infinitesimals  of  the  first  order,  and  J-  2  (qf  —  q)(V'  —  V) 
an  infinitesimal  of  the  second  order.  Hence,  putting  W  = 
—  dWq  and  W  =  dWv  ,  and  neglecting  W—W'  =  —  dWq  —  dWv 
=  J2  (?'-  q)  (V-  V\  we  have 

-dWq=dWv  (48) 

and  W"  =  -  2dWq  =  2dWv  (49) 

That  is,  during  any  infinitesimal  change  of  configuration  the 
decrease  in  the  electric  energy  of  the  system  when  the  charges 
are  kept  constant  is  equal  to  the  increase  when  the  voltages  to 
Ao  are  kept  constant  ;  and  the  energy  supplied  by  the  batteries, 
or  other  sources  of  electric  energy,  in  the  latter  case  is  equal  to 
twice  the  increase  of  electric  energy  —  one  half  going  to  increase 
the  mechanical  energy  of  the  system. 

The  principle  developed  in  this  article  will  be  applied  exten- 
sively in  what  follows  to  find  the  forcive  upon  a  given  conductor 
A  in  the  field. 

Thus  suppose  the  forcive  upon  A  to  consist  of  a  force  F  in  the 
direction  OX  of  increase  of  the  coordinate  x  of  a  point  of  A. 
Let  the  configuration  of  all  the  other  conductors  remain  fixed 
while  A  is  displaced  in  the  direction  OX  a  distance  dx.  Then 

-  dWq  =  dW9  =  Fdx 
or  F=  -  dWq$dx  =  dWJdx  (50) 

In  the  same  way,  if  the  forcive  consists  in  a  torque  T  in  the 
direction  of  an  increase  of  an  angle  0, 

T=-  dWJdO  =  dWJde  (51) 


56.  The  Discharge  by  Successive  Contacts  with  AQ  of  two  Con- 
ductors, Al  and  A^  of  the  System.     Let  all  the  other  conductors 


54  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

within  AQ  be  kept  permanently  connected  with  AQ,  thus  becoming 
permanent  parts  of  AQ.     Then  we  have  at  all  times,  by  (37)  and 

(35)'  <-  - 


Initially,  let  Al  be  insulated,  with  voltage  V  to  AQ  and  with 
charge  q,  and  let  A2  be  connected  to  AQ.  In  this  state  we  have 
from  the  above  equations 


the  first  subscript  of  ql  and  q2  denoting  the  number  of  times  the 
conductor  with  the  second  subscript  has  been  connected  to  AQ. 
Next  let  A2  be  insulated,  and  let  Al  be  then  connected  to  AQ. 
After  this  operation 


Next  let  Al  be  insulated  and  A2  then  connected  to  AQ.     Then 


Then  let  ^42  be  insulated,  and  let  Al  be  connected  to  AQ.     In 
this  state 

2^2  =  tj/^'? 


and  so  on  for  any  number  of  contacts. 

Thus  each  time  either  conductor  is  connected  to  ^0  its  charge 
is  diminished  in  the  ratio  ^122/^u^22-     After  ;z  contacts  the  charge 

of  ^  is  *    '      =  '" 


If  after  the  72th  contact  Al  is  insulated  and  A2  connected  to  AQf 
the  voltage  from  Al  to  ^40  is 

=  W/VJ"  F  (53) 


GENERAL  ELECTROSTATIC  THEORY.         55 

When  s12  is  nearly  equal  to  sn  and  s22,  which  is  the  case  when 
the  two  conductors  are  parallel  and  close  together,  especially 
when  the  dielectric  constant  of  the  medium  between  them  is 
larger  than  that  of  the  rest  of  the  dielectric  within  AQ,  the  charge 
diminishes  very  slowly  with  the  increase  of  «. 

57.  Electric  Surface  Density  and  Surface  Curvature  of  Con- 
ductors. The  electric  surface  density  upon  any  isolated  electri- 
fied conductor  is,  in  general,  greater  at  any  point  of  the  surface 
the  greater  the  curvature  at  the  point.  For  it  is  obvious  that 
the  equipotential  surfaces  drawn  about  any  such  conductor  ap- 
proach more  and  more  nearly  the  form  of  spheres  about  the 
conductor's  "  center  of  charge  "  as  their  distances  from  the  con- 
ductor increase.  That  is,  at  great  distances  the  field  is  prac- 
tically radial,  and  tubes  of  equal  strength  have  equal  cross-sec- 
tions. If  now  tubes  of  equal  strength  are  followed  backward 
toward  the  conductor,  they  become  narrower ;  and  those  which 
emanate  from  the  more  highly  curved  portions  of  the  sur- 
face become  narrower  more  rapidly  than  those  which  emanate 
from  less  highly  curved  portions,  since  the  lines  bounding 
each  tube  emanate  normally  from  the  conductor.  Thus  the 
area  from  which  a  tube  of  given  strength  starts  is  smaller,  or 
the  electric  surface  density  upon  it  greater,  the  more  highly  the 
surface  is  curved  (convex  outward)  at  the  point. 

In  the  same  manner  the  density  is  smaller  the  more  concave 
the  surface. 

At  a  sharp  edge  or  point  the  density  is  very  great.  If  the 
edge  or  point  were  really  sharp,  the  density  there  would,  of 
course,  be  infinite,  as  tubes  would  emanate  from  bases  of  no 
dimensions. 

Where  two  parts  of  a  conducting  surface  make  with  one  an- 
other a  reentrant  angle  the  surface  density  vanishes,  since  any 
displacement  there  would  be  perpendicular  to  both  parts  of  the 
surface,  which  is  impossible. 

In  addition  to  the  curvature  at  the  given  point,  the  curvature 
of  the  neighboring  parts  of  the  surface  is  of  importance  in  deter- 


56  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

mining  the  density.  Thus  the  density  would  be  small  on  a 
"point  "  on  the  inside  of  a  vessel  nearly  closed,  and  it  might  be 
great  in  a  small  cavity  in  a  highly  convex  portion  of  the  outer 
surface. 

If  the  conductor  is  not  isolated,  the  effects  here  described  will 
be  rendered  more  or  less  conspicuous  according  to  the  signs, 
magnitudes,  and  distributions  of  the  charges  on  neighboring 
bodies. 

In  the  following  chapter  many  examples  illustrating  the  prin- 
ciple of  this  article  will  be  found. 

58,  The  Capacity  of  a  Conductor  is  a  very  commonly  used 
and  convenient,  but  otherwise  objectionable,  abbreviation  for  the 
permittance  of  the  dielectric  (supposed  homogeneous  and  iso- 
tropic)  enclosed  between  the  conductor  and  an  infinitely  remote 
surrounding  conductor  when  no  other  conductors  or  electrified 
bodies  are  present. 


CHAPTER    II. 

SIMPLE    IDEAL    ELECTRIC    FIELDS  AND   CONDENSERS  WITH 
HOMOGENEOUS    DIELECTRICS. 

In  the  following  articles  describing  various  electric  fields  all 
more  or  less  ideal,  the  dielectric  is  supposed  to  be  homogeneous 
and  isotropic  throughout,  and  the  electrified  bodies  in  each  case 
are  supposed  to  be  infinitely  remote  from  all  other  electrified 
bodies,  unless  the  contrary  is  stated.  The  potential  at  a  point 
will  be  taken  in  this  chapter  and  in  Chapter  IV.  as  the  line  inte- 
gral of  the  electric  intensity  from  the  point  to  a  region  infinitely 
distant  from  all  electrified  bodies. 

1.  The  Spherically  Radial  Electric  Field.  Let  an  electric 
charge  q  be  concentrated  at  a  point  P.  The  field  can  be  found 
at  once  from  (i)  and  (2),  Chapter  I.,  or  from  Gauss's  theorem 
and  the  principle  of  symmetry.  By  symmetry,  the  electric  dis- 
placement is  directed  radially  from  P  (or  to  P  if  g  is  negative) 
and  has  the  same  magnitude  at  eveiy  point  of  any  sphere  with 
center  at  P.  All  such  spheres  are  evidently  equipotential  sur- 
faces. Since  the  electric  flux  across  any  of  these  equipotentials 
is  q,  the  flux  per  unit  area  across  a  sphere  of  radius  L,  or  the 
electric  displacement  at  a  distance  L  from  P,  is 


(I) 
from  which 


(2) 
From  (2),  the  potential  at  a  point  distant  L  from  P  is 


=    f 

J  L 


L 

57 


(3) 


58  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

the  integration  being  performed  along  a  line  of  intensity  for  sim- 
plicity. 

Maxwell's  plane  diagram  of  the  field  is  given  in  Fig.  14  and 
described  in  §  7. 

2,  The  Spherical  Condenser.  If  in  the  radial  field  of  §  i  infi- 
nitely thin  conducting  sheets  are  placed  coincident  with  two 
equipotential  spheres  of  radii  L^  and  L2  =  Lv  -J-  d,  the  electric 
field  will  remain  unaltered,  except  that  it  will  be  rendered  dis- 
continuous at  the  surfaces  of  the  conducting  sheet  (§  47, 
Chapter  I.).  The  charge  upon  the  outer  surface  of  the  inner 
sphere  is  now  q,  and  that  upon  the  inner  surface  of  the  outer 
sphere  is  —  qy  the  two  surfaces  with  the  intervening  dielec- 
tric forming  a  condenser  whose  field  is  radial  and  given  by  (i) 
and  (2). 

The  charge  q  at  the  center  of  the  spheres  is  the  electric  image 
in  the  inner  sphere  of  the  charge  on  the  inner  surface  of  the  outer 
sphere  and  all  external  charges ;  or,  if  one  of  the  conducting 
sheets  is  removed,  q  is  the  electric  image  in  the  remaining  sphere 
of  the  complementary  charges  at  an  infinite  distance  (on  the  in- 
finite sphere  at  zero  potential  surrounding  q).  The  conducting 
substance  may  be  extended  into  the  regions  within  the  inner 
sphere  and  without  the  outer  sphere  in  any  manner,  or  the  fields 
in  these  regions  may  be  wholly  destroyed,  without  affecting  the 
field  of  the  condenser. 

For  the  voltage  between  the  two  conductors  (2)  gives 

V,-V2=    r^L=g/47rc.(i/Ll-i/L2)  =  ^/47rcLlL2    (4) 
JL, 

The  capacity  of  the  condenser  is 

•S  =  ?/( ^i  -  ^)  =  Vr^M-  4^L^jd-(i  +  <//£,)     (5) 
and  the  capacity  per  unit  area  of  the  inner  sphere  is 

S'  =  S/47T/V2  =  cjd-  (i  +  djL,)  (6) 


ELECTRIC   FIELDS   AND    CONDENSERS.  59 

The  energy  contained  in  the  dielectric,  or  the  energy  of  the 
condenser,  is 


W=  \q(  V,  -  F2)  =  fdltorcLft.  +  djL,)  =  , 

d-  (  i  +  djL,)  -(Vi- 


Limiting  Cases,  (i)  The  parallel  plate  condenser.  If  d  is 
kept  constant,  and  L  made  to  increase,  the  electric  field  normal 
to  a  given  portion  of  the  inner  or  outer  sphere  obviously  ap- 
proaches uniformity.  When  L  becomes  infinite,  any  finite  por- 
tion of  the  condenser  becomes  a  parallel  plate  condenser  (§  12) 
of  capacity  per  unit  area  S'  =  cjd.  In  any  case  when  djL  is 
small,  the  field  is  approximately  uniform  (in  magnitude)  and  the 
capacity  per  unit  area  approximately  cjd.  This  field  is  fully 
discussed  in  §  12. 

(2)  The  isolated  sphere.  If  Z2  is  made  infinite  while  L  re- 
mains constant,  (4),  (5),  (6),  and  (7)  become 

(8) 
(9) 

V-'/A  (10) 

and 

21TCL  i  i 


The  coefficients  of  potential  and  capacity  for  the  system  of 
two  spheres  can  be  easily  found  from  the  equations  of  §  50,  I., 
together  with  those  just  developed.  Thus 


Ai  =  ^i  M  (<72  =  o) 
V,  =  o)  =  -  5  =  -  su 


V\  =  °)=  S  +    S2  ( 


6o 


ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 


3.  Laplace's  Equation  for  the  Spherically  Radial  Field.  As 
an  example  of  the  use  of  Laplace's  equation  we  will  determine 
Vl  —  V2  by  a  different  method.  Since  the  field  is  radial,  the 
equation  may,  with  the  aid  of  Fig.  12,  be  put  in  a  much  simpler 
form  than  that  of  (21),  Chapter  I.  The  simplified  form  could  be 
obtained  from  (21),  Chapter  I.,  by  a  mathematical  transformation, 
the  proper  conditions  being  put  in,  but  can  be  developed  more 
simply  by  starting  from  first  principles. 


The  electric  flux  into  the  elementary  volume  dr  across  the 
surface  dS  is  DdS.  The  flux  out  from  the  volume  across  dSf  is 
(D  -f-  dDjdL- dL)dSf ,  and  there  is  no  flux  across  any  other  part 
of  the  tube.  Hence  the  resultant  flux  outward  is 

D(dSr  -  dS)  -f  dDjdL  •  dLdS  =  pdLdS  =  pdr  =  o 

Dividing  this  equation  by  dS,  writing  for  dS' jdS  its  equal 
(L  -f  dL)2/L2,  and  passing  to  the  limit,  we  obtain,  on  putting 
D  =  cE  equal  to  —  cdVfdL, 

which  is  the  form  taken  by  Laplace's  equation  for  a  radial  field. 
From  (13)  we  obtain  by  integration 


DdVjdL  =  Cv  or  dVjdL  =  CJD 
where  £\  is  a  constant  to  be  determined,  and 


(a) 


ELECTRIC    FIELDS   AND    CONDENSERS.  6  1 

Integrating  from  L  =  .Ll  to  L  =  L2,  we  have 


Since  when  L  =  Lv  d  VjdL  =  —  o-fc,  (a)  gives 

£--.-»Ay* 

Hence  (<r)  becomes 


?/4«  •(1/4- 

which  is  identical  with  (4). 

The  potential  at  any  point  distant  L,  greater  than  ZT  and  less 
than  L2  (the  field  being  confined  within  the  space  between  the 
spheres  of  radii  Z:  and  -Z2)  from  P  is 

F=  V,  +  C,  f^L/L2  (14) 

JL{ 

4.  The  Potential  at  a  Point  Due  to  Any  Electric  Distribution  in 
a  Homogeneous  Isotropic  Dielectric.  For  the  potential  at  a  point 
distant  Lv  L2,  •  •  •  ,  Ln  from  point  charges  qv  q2J  •  •  •  ,  qn,  respec- 
tively, §  33,  Chapter  L,  gives,  by  means  of  (3), 

V=  I/C47T  •  (qi  IL,  +  qJL2  +  .  .  -  +  ?JLn)  =  I  fair  ^qjL  (l  5) 


If  the  charges,  instead  of  being  concentrated  at  points,  which, 
to  be  exact,  is  of  course  impossible,  are  distributed  over  surfaces 
and  through  volumes,  (15)  becomes 

S/Z  +  fpdrjL}          (16) 

the  first  integration  extending  over  all  electrified  surfaces,  and 
the  second  throughout  all  electrified  volumes. 

While  (15)  and  (16)  have  been  deduced  for  a  space  filled  up 
with  a  single  dielectric,  they  are  also  true,  by  §  28,  Chapter  L, 
when  the  field  contains  any  number  of  conductors.  The  equa- 
tions will  be  extended  later  to  include  all  cases  (IV.). 

5,  The  Law  of  Inverse  Squares.  A  consideration  of  equations 
(l)  and  (2)  shows  that  the  law  of  inverse  squares,  which  they 


62  ELEMENTS    OF   ELECTROMAGNETIC   THEORY. 

state  in  its  simplest  form,  is  due  to  the  continuity  of  the  electric 
displacement  (or  the  "  incompressibility  of  electricity"),  the  flux 
from  a  charge  q  being  q  across  every  surface  surrounding  the 
charge,  and  to  the  spherical  or  three-dimensional  nature  of  space, 
the  flux  from  a  point  charge  being  distributed  equally  in  all  direc- 
tions (when  the  medium  is  homogeneous  and  isotropic,  in  which 
case  only  the  law  is  valid). 

6.  The  Normal  Electric  Field  and  the  Potential  of  the  Earth. 

Numerous  investigations  upon  the  electrical  state  of  the  earth's 
atmosphere,  made  at  altitudes  above  its  surface  ranging  from 
nothing  up  to  4000  meters,  have  shown  that  the  atmosphere  is 
the  seat  of  an  electric  field  whose  intensity,  in  normal  conditions, 
is  directed  toward  the  earth. 

In  good  weather,  the  magnitude  of  this  intensity  at  the  earth's 
surface  ranges  from  about  0.00005  RES  unit  (about  50  volts/ 
meter)  to  about  0.00040  RES  unit  (about  400  volts/meter),  ac- 
cording to  season,  locality,  etc.  Thus  the  electric  surface  density 
of  the  earth's  surface,  in  normal  weather,  ranges  from  about 
—0.00005  RES  unit  to  about  —  0.00040  RES  unit.  The  magni- 
tude of  the  intensity  increases  with  the  altitude  above  the  earth's 
surface  up  to  heights  of  some  2000  meters,  showing  that  the 
atmosphere  in  this  region,  like  the  earth's  surface,  is  negatively 
charged.  In  the  higher  regions  of  the  atmosphere,  on  the  other 
hand,  the  intensity  decreases  with  the  increase  of  altitude,  with- 
out becoming  greatly  reduced,  however,  at  the  greatest  altitudes 
yet  investigated.  Thus  the  higher  regions  of  the  atmosphere  are 
positively  charged  ;  but  whether  all  the  tubes  of  displacement 
terminating  upon  the  earth  and  in  the  lower  regions  of  the  at^ 
mosphere  originate  in  the  upper  regions,  or  whether  some  of  these 
tubes  emanate  from  other  bodies  in  space,  is  not  yet  known.  If 
further  investigation  demonstrates  that  at  greater  altitudes  the 
intensity  vanishes,  the  former  alternative  will  be  shown  to  be 
correct.  The  altitudes  here  considered  are  so  small  that  no  sen- 
sible variation  in  the  intensity  would  occur  within  them  owing  to 


ELECTRIC    FIELDS    AND    CONDENSERS.  63 

the  increase  in  the  cross-section  of  the  tubes  of  induction  with 
the  distance  from  the  surface  of  the  earth. 

The  earth  itself  in  any  case  is  negatively  charged  ;  and  since 
the  electric  intensity  in  the  atmosphere  is  directed  toward  the 
earth,  its  potential  is  negative  if  the  (wholly  unknown)  electrifi- 
cation of  all  other  bodies  in  space  beyond  the  atmosphere  is  not 
considered.  From  the  magnitude  of  the  intensity  given  above  and 
from  the  great  altitude  to  which  the  field  extends  without  great 
diminution  in  strength,  it  is  obvious  that  the  magnitude  of  this 
potential  is  very  great. 

It  follows  from  (15)  that  that  part  of  the  potential  at  the  center 
of  a  conducting  sphere  of  radius  L  due  to  any  charges  2</  upon 
its  surface  is  ^ql^-cL.  Since  the  sphere  is  conducting,  this  ex- 
pression gives  the  part  of  the  potential  at  any  point  of  the  sphere 
due  to  the  surface  distribution.  From  this  and  §  6,  L,  it  follows 
that  the  potential  of  the  earth  is  not  appreciably  affected  by  the 
the  development  of  any  charges  retained  upon  or  near  its  sur- 
face. For  by  §  6,  L,  2^  is  always  zero  ;  and  Z,  the  distance  from 
the  center  of  the  earth,  is  very  great  and  practically  the  same 
for  all  the  charges. 

The  field  surrounding  the  earth,  as  a  matter  of  fact,  is  by  no 
means  strictly  static,  and  the  surface  of  the  earth  is  never  strictly 
an  equipotential. 

7.  Maxwell's  Plane  Diagram  of  the  Spherically  Radial  Field, 

Maxwell's  diagrams  are  all  so  drawn  that  the  successive  equipo- 
tential surfaces  differ  in  potential  by  the  same  amount  (for  ex- 
ample, unity),  and  that  the  tubes  of  induction  corresponding  to 
the  intervals  in  the  diagram  between  successive  lines  of  displace- 
ment are  of  equal  strength  (for  example,  unit  tubes). 

(  i  )  The  equipotential  surfaces.  The  radius  of  the  sphere  whose 
potential  is  Fis 

L  = 


Hence  by  giving   Fin  succession  the  values  I,  2,  3,  etc.,  the 
radii  of  the  equipotentials  with  these  values  of  the  potential  can 


64 


ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 


be  obtained.  The  circles  in  which  any  plane  passing  through  the 
charge  cuts  these  spheres  form  the  equipotential  lines  in  the 
diagram.  The  circles  may  of  course  be  drawn  for  any  constant 
increment  of  potential  instead  of  unity. 


Fig.  13. 

(2)  The  lines  and  tubes  of  displacement.  The  lines  of  displace- 
ment are  straight  lines  radiating  from  the  charge.  The  tubes  of 
displacement  in  Maxwell's  method  are  formed  by  rotating  the 
diagram  of  lines  of  displacement  about  a  straight  line  drawn 
through  the  charge.  We  proceed  to  find  the  distribution  of  the 
lines  in  the  plane  diagram  when  drawn  so  that  the  tubes  thus 
formed  are  of  equal  strength. 

Let  a  circle  of  any  radius  AP,  Fig.  13,  be  drawn  about  P,  the 
seat  of  the  charge,  as  center ;  and  let  the  diameter  AB  be  divided 
up  into  q  equal  parts  by  straight  lines  drawn  perpendicular  to  AB 
and  cutting  the  circle  in  the  points  n,  22,  etc.  From  P  let 
straight  lines  be  drawn  through  1 1,  22,  33,  etc.  These  lines  are 
the  lines  of  Maxwell's  diagram. 

For  if  the  figure  is  rotated  about  AB  as  axis,  the  circle  traces 
out  a  sphere,  the  lines  u,  22,  etc.,  trace  out  equidistant  parallel 


ELECTRIC    FIELDS    AND    CONDENSERS.  65 

planes  which  cut  the  sphere  up  into  zones  lAi,  21 12,  etc.,  of  equal 
area  (ijq  that  of  the  sphere).  Hence  the  lines  Pi,  P2,  and  P$% 
etc.,  trace  out  cones  iPi,  iP2,  2P$,  etc.,  through  each  of  which 
the  electric  flux  is  the  same  and  equal  to  I  jq  x  q  =  unity.  Hence 
these  cones  are  the  tubes  of  displacement  required,  and  the  lines 
Pi,  P2,  P$,  etc.,  are  the  lines  of  displacement  in  the  diagram. 

The  field  may  of  course  be  divided  up  into  tubes  of  any  other 
strength  instead  of  unity  by  cutting  AB  up  into  the  desired 
number  of  equal  parts  in  the  above  construction. 

The  diagram  is  given  in  Fig.  14  for  the  case  in  which  the 
strength  of  each  tube  is  taken  as  q/S. 


Fig.  14. 

8.  The  Cylindrically  Radial  Field,  or  field  surrounding  a  uni- 
formly electrified  infinite  straight  line  or  circular  cylinder.  Con- 
sider first  an  electrified  straight  line,  and  let  the  charge  on  unit 
length  be  denoted  by  q.  By  symmetiy  D  is  everywhere  normal 
to  the  line  and  to  the  circular  cylinders  about  it  as  axis,  which 
are  the  equipotential  surfaces,  and  has  the  same  magnitude  at 
every  point  of  any  such  cylinder.  Since  the  flux  across  a  length 
A  of  any  equipotential  is  qA,  the  flux  per  unit  area  across  a  cylin- 


66  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

der  of  radius  Z,  or  the  electric  displacement  at  distance  L  from 

the  axis,  is 

D  =  qAJ2TrLA  =  q  \2irL  (17) 

whence  E=  Djc  =  q  j  2ircL  (18) 

The  potential  at  a  point  distant  L  from  the  axis  is 

V=ql27rcf  dLjL  (19) 

JL 

9.  The  Cylindrical  Condenser.  If  two  equipotentials  of  radii 
L^  and  L2  =  L^  -f  d  are  replaced  by  infinitely  thin  conductors, 
the  electric  field  will  remain  unaltered  except  that  it  will  become 
discontinuous  at  the  surfaces  of  the  conductors.  The  charge 
upon  unit  length  of  the  outer  surface  of  the  inner  cylinder  is  now 
q,  and  that  upon  unit  length  of  the  inner  surface  of  the  outer 
cylinder  is  —  qy  and  the  two  conducting  surfaces  with  the  inter- 
vening dielectric  form  a  condenser  whose  field  is  given  by  (17) 
and  (18).  The  charge  upon  the  straight  line  and  that  on  the 
inner  surface  of  the  outer  cylinder  together  with  all  the  external 
charges  are  electric  images  of  one  another  in  the  inner  cylinder, 
etc.  The  conducting  substance  may  be  extended  into  the 
regions  within  the  inner  surface  and  without  the  outer  surface  in 
any  manner,  or  the  fields  of  these  regions  may  be  wholly  de- 
stroyed, without  affecting  the  field  of  the  condenser. 

For  the  voltage  between  the  plates  of  the  condenser,  (18)  gives 


F'ttL/L  =  ql2irc  -  log(l  +  djL^      (20) 
JL-, 


-v,= 

^ii 
The  capacity  of  a  length  A  of  the  condenser  is 

and  the  capacity  per  unit  area  of  the  inner  cylinder  is 

Sf  =  S/27rL,A  =  cIL  log  (i  -f  d!L\ 

(22) 


ELECTRIC    FIELDS    AND    CONDENSERS.  67 

If  d  is  kept  constant  and  Z1  made  to  increase,  the  field  normal 
to  a  given  portion  of  the  inner  (or  outer)  cylinder  evidently  ap- 
proaches uniformity  ;  and  in  the  limit,  when  Zx  =  infinity,  any 
finite  portion  of  the  condenser  becomes  a  parallel  plate  condenser 
(§  12)  of  capacity  cfd  per  unit  area.  In  any  case  when  djL  is 
small  the  capacity  per  unit  area  is  approximately  cjd. 

The  energy  of  a  length  A  of  the  condenser  is 


W=  \qA(V,  -  F2)  =  fA\4*c.  log  (i 


The  field  of  an  infinite  isolated  circular  cylinder  uniformly 
charged  is  given  by  the  above  equations  on  making  L2  infinite 
and  V2  zero. 

Vl  —  V^  can  be  easily  obtained  by  the  direct  application  of  the 
law  of  inverse  squares.  Let  the  field  outside  the  condenser  be 
zero  (though  the  results  obtained  will  be  independent  of  this 
assumption);  then  V2  =  o,  and  V^  is  the  potential  at  any  point 
on  or  within  the  inner  cylinder,  and  is  therefore  the  potential  at 
any  point  P  on  this  axis.  Hence,  from  the  figure  (Fig.  1  5), 


1 

1 

''-''     \  \ 

L^ 

,X>""'tU 

""*        ii 

~                                                     r. 

T*     ~ 

f\ 

*    fv 

Fig.  15. 


-  V2  =  V,  =  2  r^TrL^/^c^  +  A2)* 
J<> 


(a) 
log  LJL^ 

as  in  (20)  above. 

10.    Laplace's   Equation   for   the   Cylindrically  Eadial   Field. 
Vl  —  V2  can  also  be  obtained  directly  from   Laplace's  equation. 


68  ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

Without  transforming  the  general  equation,  we  can  obtain 
directly,  by  a  simple  process  similar  to  that  employed  in  §  3,  the 
special  form  it  assumes  in  a  cylindrically  radial  field.  Thus  we 

find 

Ld2  V\dL*  +  dVjdL  =  o  (24) 

Hence,  by  integration, 

LdVldL  =  C] 
or  (a) 


and 

V,  -  V,  =  C  ridLjL  =  gl27rc   log  Z2/ZX  (b) 

JL* 

since  when  L  =  Lv  (a)  gives 

C=  -Ll<rjc=  -  .gl2>jrc  (c} 

The  potential  at  any  point  between  the  two  cylinders,  distant 
L  from  the  axis,  is 


JL 


11.  Maxwell's  Plane  Diagram  of  the  Cylindrically  Radial  Field. 
This  diagram,  like  that  of  §  7,  is  drawn  so  that  the  tubes  of  dis- 
placement corresponding  to  the  intervals  between  the  successive 
lines  of  displacement  are  of  equal  strength,  and  so  that  the  volt- 
age between  successive  equipotential  lines  or  surfaces  is  constant. 

Since  every  line  of  displacement  lies  wholly  in  a  plane  perpen- 
dicular to  the  axis  of  the  cylinders  or  electrified  straight  line,  and 
since  the  lines  of  displacement  are  exactly  similar  in  every  such 
plane,  any  such  plane  is  chosen  as  the  plane  of  the  diagram,  and 
the  tubes  of  displacement  are  supposed  to  be  formed  by  moving 
the  diagram  perpendicularly  to  its  plane.  We  shall  suppose  the 
diagram  to  represent  unit  depth  of  the  field,  all  the  tubes  having 
this  thickness. 

I.  The  equipotential  lines  in  the  diagram  are  circles  centered 
on  the  axis.  Though  the  potential  of  every  circle,  as  given  by 


ELECTRIC    FIELDS   AND    CONDENSERS.  69 

(19),  is  infinite,  we  may  draw  a  system  of  circles  differing  in 
potential  successively  by  a  constant  finite  quantity,  as  unity,  by 
starting  with  any  equipotential  circle  of  any  radius  a  and  potential 


Fig.   16. 


Va  (infinite)  and  applying  (20)  to  find  the  radius  L  of  the  circle 
whose  potential  is  Va  —  VL  less  than  Va.     Thus  we  have 


By  giving  to  Va  —  VL  in  succession  the  values  I,  2,  3,  etc.  (or 
any  set  of  successive  values  differing  by  a  constant),  as  many 
circles  of  the  system  as  desired  may  be  obtained. 

2.  The  lines  of  displacement  are  straight  lines  drawn  from  the 
center  of  the  circles  and  dividing  each  circle  into  q  (or  any  inte- 
gral number)  of  equal  parts. 

Such  a  diagram  is  shown  in  Fig.  16. 

12.  The  Uniform  Electric  Field.  Let  the  field  be  terminated 
by  an  infinite  plane  conducting  surface.  The  surface  will  be 
uniformly  electrified,  the  displacement  everywhere  uniform  and 
normal  to  the  surface,  and  the  equipotentials  planes  parallel  to 
the  surface.  The  displacement  is 


70  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

D=(T  (26) 

and  the  intensity  is         E=  Djc  =  a/c  (27) 

The  Parallel  Plate  Condenser.  If  an  infinitely  thin  conduct- 
ing sheet  is  placed  coincident  with  the  equipotential  plane  dis- 
tant d  from  the  electrified  surface,  the  electric  field  will  remain 
unaltered  except  for  the  discontinuity  introduced  at  the  surfaces 
of  the  sheet,  and  each  side  of  the  sheet  will  therefore  have  the 
same  electric  surface  density  o-  (numerically).  The  two  adja- 
cent surfaces  and  the  dielectric  between  them  form  a  condenser 
whose  field  is  uniform  and  given  by  (26)  and  (27),  and  which 
will  remain  unaltered  if  the  conductors  are  extended  into  the 
region  outside  the  condenser. 

The  voltage  between  the  two  conductors  is 

V,-  V2  =  Ed=(Tdjc  (28) 

and  the  capacity  of  a  portion  of  the  condenser  of  right  cross- 
section  A  is 

5  =  q\(  Vl  -  F2)  =AjEd  =  AD  I  Ed  =  Acjd  (29) 

The  capacity  per  unit  area  is,  as  already  proved  less  directly 

in  §§2  and  9)  Sf~SfA=cfd  (30) 

The  energy  of  a  portion  of  the  condenser  of  right  cross- 
section  A  is 


=  \Ac\d  -(V,-  Vtf  =  \EDAd    (31) 

The  force  F  (positive  when  tending  to  increase  d,  or  to  separate 
the  plates)  upon  an  area  A  of  either  plate,  if  the   tubes   in   the 
condenser  are  the  only  tubes  terminating  upon  the  plates  (that 
is,  if  there  is  no  external  field),  is 
F=  —  dW\dd(<r  constant)  =  -  \<^A\c  =  -  \cE2A  =  etc. 

=  +  dWjdd  [  (  Vl  -  F2)  constant]  (32) 

=  _  %AC  (V,  -  F2)2/^2  =  -\cE*A  =  etc. 
by  §  55,  Chapter  I. 


ELECTRIC    FIELDS   AND    CONDENSERS.  7  1 

This  result  also  follows  from  §  40,  I.,  or  the  result  there  given 
follows  from  (32). 

Vv  —  V.2  can  be  obtained  also  by  the  direct  application  of  the 
law  of  inverse  squares.  We  assume  that  there  is  no  field  ex- 
ternal to  the  region  within  the  plates,  though  the  results  will  of 
course  be  independent  of  this  assumption,  since  the  internal  and 
external  fields  are  wholly  independent  of  one  another.  From 
any  point  P  of  the  positive  plate  imagine  a  straight  line  drawn  per- 
pendicular to  both  plates.  Imagine  the  surfaces  of  the  conduc- 
tors divided  up  into  infinitesimal  circular  zones  centered  on  this 
line,  and  let  x  denote  the  radius  of  any  zone  and  dx  its  width. 
Then  the  potential  at  P,  i.  e.t  the  potential  at  all  points  of  the 
positive  plate,  is  evidently 

V^  =    I      [—  <T2rjrxdxl^irc(x'L  -f  d  )*  -f-  aZirxdxlqjrcx} 

Jo 

=  <r/2c  f    [I  -  xj(x2  +  d*)*\dx=  +  <rdJ2c 
Similarly  V2  =  —  <rd/2c 

Hence  Vl  -  V2  =  <rdjc  (28) 

Laplace's  Equation  for  a  Uniform  Field,  The  same  result  can 
be  obtained  also  from  Laplace's  equation.  In  a  uniform  field,  if 
we  take  X  in  the  direction  of  Dy  the  equation  (21),  Chapter  I., 
simplifies  to 


=  o  (33) 

since  £>2  =  £>3  =  o  (or  dV\dy  =  dVjdz  =  o). 
By  integration,  (33)  gives 

dVldx=Ci=-<rlc  (a) 

By  a  second  integration 

V^  -  V2  =  C,(o  -  d}  =  -  CJ  =  vdjc  (28) 

At  any  point  distant  x  from  the  positive  plate,  when  x  is  less 
than  d,  we  have 

V=  V,  +  Crr  (34) 


ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 


13,  Maxwell's  Plane  Diagrams  of  the  Uniform  Field,  (i)  The 
equipotentials  are  equidistant  planes  perpendicular  to  the  lines  of 
displacement. 

(2)  The  diagram  of  the  tubes  of  displacement  may  be  drawn  in 
two  different  ways.  If  the  tubes  are  to  be  formed  by  moving 


Fig.  17. 

the  diagram  perpendicularly  to  its  own  plane,  the  corresponding 
lines  in  the  diagram  must  be  drawn  equidistant.  But  if  the  tubes 
are  to  be  mapped  out  by  rotating  the  diagram  about  a  line  of 
displacement  as  axis,  the  distances  of  the  successive  lines  from 
the  axis  (or  the  radii  of  the  outer  surfaces  of  the  cylindrical 
tubes)  may  be  found  by  giving  to  the  expression  TrR2D(  =  the 
flux  through  a  tube  of  radius  R)  values  which  are  multiples  of 
the  successive  whole  numbers  by  a  constant,  and  solving  for  the 
corresponding  values  of  R.  A  diagram  of  the  former  kind  is 
given  in  Fig.  1 7,  and  one  of  the  latter  kind  in  Fig.  1 8. 


Fig.   18. 


14.  Maxwell's  Plane  Diagram  of  the  Resultant  of  two  Fields. 

If  the  plane  diagrams  of  two  fields  are  given,  both  drawn  for  the 
same  strength  of  tubes  and  the  same  potential  differences,  and  if 


ELECTRIC   FIELDS    AND    CONDENSERS. 


73 


both  are  diagrams  which  trace  out  the  tubes  of  displacement  by 
their  revolution  about  the  same  axis,  or  by  motion  at  right 
angles  to  their  plane  through  the  same  distance,  the  resultant 
diagram  of  the  two  fields  superposed  can  easily  be  drawn. 

( i )  The  equipotentials.  Since  the  potential  of  each  line  in  both 
diagrams  is  known,  the  potential  of  every  point  of  intersection 
when  the  diagrams  are  superposed  is  known.  Hence  by  draw- 
ing curves  through  all  the  points  of  intersection  which  have  the 
same  potential  we  get  the  resultant  equipotential  curves.  This 
is  equivalent  to  drawing  the  curves  forming  the  diagonals  of  the 


quadrilaterals  made  by  the  superposition  of  the  two  systems  of 
equipotentials,  since  in  passing  from  one  corner  to  the  other  the 
potential  of  one  diagram  diminishes  as  much  as  that  of  the  other 
increases.  There  is  no  difficulty  in  choosing  the  proper  diagonal. 
The  difference  of  potential  between  the  successive  curves  in  the 
resultant  diagram  is  the  same  as  that  between  the  successive 
curves  in  the  original  diagrams.  A  particular  case  is  illustrated 
in  Fig.  1 9,  the  lines  of  the  resultant  diagram  being  dotted. 

(2)  The  lines  of  displacement.  The  lines  of  displacement  in 
the  resultant  diagram  are  the  curves  forming  the  diagonals  of 
the  quadrilaterals  resulting  from  the  superposition  of  the  two 


74 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


diagrams  of  lines  of  displacement.  For  across  every  element  of 
such  a  curve,  as  ab  in  the  figure  (Fig.  20),  the  flux  is  zero,  since 
the  flux  through  one  tube  as  T in  one  direction  just  cancels  that 
through  another  as  T'  in  the  other  direction.  It  is  also  obvious 


Fig.  20. 

from  the  figure  that  the  flux  along  any  resultant  tube  is  equal  to 
that  along  any  of  the  original  tubes.  No  difficulty  can  be  ex- 
perienced in  choosing  the  proper  corners  of  the  quadrilaterals  to 
connect. 

By  compounding  diagrams  in  pairs  it  is  clear  that  the  diagram 
of  the  resultant  of  any  number  of  fields  superposed  can  be  ob- 
tained by  the  above  method. 

15,  The  Field  Terminated  by  two  Equal  and  Opposite  Concen- 
trated Charges.  Let  the  charges,  which  will  be  denoted  by  q 
and  —  qy  be  located  at  A  and  B,  Fig.  2 1 ,  distant  2d  apart.  The 
field  is  evidently  symmetrical  about  the  line  AB,  and  is  the  re- 
sultant of  two  radial  fields,  §  I .  The  displacement  and  intensity 
at  any  point  P  distant  Ll  from  A  and  L2  from  B  are  therefore 


D  =  Vector  sum  of 


and 


At- 


directed  from  A)  and 
directed  toward  B) 


For  the  potential  at  P,  (36)  gives 


(35) 
(36) 

(37) 


ELECTRIC    FIELDS   AND    CONDENSERS. 


75 


(37)  is  also  the  equation  of  the  equipotential  surface  whose 
potential  is  V. 

The  lines  of  intensity  are  the  lines  orthogonal  to  the  surfaces 
given  by  (37).  The  equation  of  a  line  of  intensity  can  be  ob- 
tained at  once  by  writing  down  the  condition  that  there  is  no 


E2 


component  of  electric  intensity  perpendicular  to  such  a  line.  If 
ds  (Fig.  21)  is  an  element  at  P  of  the  line  of  intensity  through 
P,  and  if  a^  and  a2  are  the  angles  made  by  El  and  E2  with  the 
normal  N  at  P,  we  have,  to  express  this  condition, 

El  cos  ax  -f  E2  cos  a2  =  o 

If  Zj  and  L2  make  with  AB  the  angles  6l  and  #2,  this  condition 
may  be  written,  as  the  figure  shows, 


Multiplying  by  PD,  the  perpendicular  to  AB  from  P,  and  inte- 
grating, we  have 

Constant  =  C=  —  (/sin  6ld6l  +/sin  02d6^=  cos  6^+  cos  02  (38) 


which  is  the  equation  sought,  in  terms  of  Ol  and  02. 

By  giving  to  C  different  values,  the  equation  of  any  line  of 
intensity  may  be  obtained.     To   find   C  for  the  line  which  cuts 


76  ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

the  plane  normal  to  AB  through  its  central  point  C  at  a  distance 
x  from  C,  we  have,  for  this  point,  6l  =  02,  and 


Therefore 

C=  2  cos  0l=2  cos  0a  =  2^/(^/2  +  **}  (39) 


The  lines  of  displacement  and  the  equipotential  lines,  drawn 
by  the  method  of  Maxwell,  §  14,  are  shown  in  Fig.  22.  (See 
(Maxwell's  Treatise,  §123.) 

If  the  distance  2d  is  diminished  indefinitely  and  the  charges  q 
and  —  q  increased  in  such  a  way  that  q  x  2d  =  constant  =  M, 
the  system  becomes  a  point  doublet  of  moment  M.  This  doublet 
and  its  field  are  discussed  in  §  27. 

From  (37)  it  follows  that  the  infinite  plane  perpendicular  to 
AB  through  its  middle  point  Cis  at  zero  potential. 

At  any  point  on  this  plane  the  resultant  intensity  and  displace- 
ment are  normal  in  the  direction  AB.  If  P  is  distant  x  from  C, 
the  displacement  at  P  will  be,  by  (35), 

D  =  2ql4ir(d2  +  xz]  -  dj(d2  +  x^  =  qdl2ir(dz  +  x^    (40) 

If  for  the  infinite  plane  equipotential  surface  through  C  an  in- 
finitely thin  conducting  sheet  is  substituted,  the  field  on  either 
side  will  remain  unaltered,  and  the  two  point  charges  will  be- 
come electric  images  of  one  another  in  the  sheet. 

If  the  field  on  the  side  toward  B  is  destroyed,  or  if  the  con- 
ductor is  extended  toward  B  in  any  manner,  the  field  on  the  side 
toward  A  will  not  be  affected,  and  we  have  the  electric  field 
bounded  by  a  concentrated  charge  q  and  the  (induced)  charge 
upon  an  infinite  plane  conducting  surface  distant  d  from  q  and 
maintained  at  zero  potential.  If  q  is  positive  the  electric  surface 
density  is  negative  at  every  point  of  the  surface,  since  D  there 
has  the  direction  AB.  The  magnitude  of  D  =  a  is  given  in 
(40).  The  total  charge  upon  the  infinite  plane  is  —  q,  since  all 
the  tubes  from  A  terminate  upon  the  plane. 


ELECTRIC    FIELDS    AND    CONDENSERS. 


77 


Since  the  field  about  A  was  unaltered  by  the  introduction  of 
the  conducting  sheet  and  the  destruction  of  the  field  on  the  side 
toward  B,  the  force  between  the  charged  body  at  A  and  the  plane 
conductor  is  the  same  as  the  force  formerly  acting  between  A 

and  B.     That  is, 

F =  -  q2 1 16-rrcd2  (41) 

Since  concentrated  charges  do  not  exist,  we  shall  suppose  the 
charges  at  A  and  B  distributed  over  extremely  small  conducting 


Fig.  22. 


spheres  each  of  radius  a,  so  that  the  field  in  the  region  outside 
the  spheres  will  be  practically  the  same  as  that  already  discussed. 
From  (34),  I.,  the  potentials  of  the  spheres  A  and  B  are 


-        and 


i2-  22 

the  subscripts  i  and  2  being  applied  to  A  and  B  respectively. 
Since  the  spheres  are  very  small,  these  equations  become  ver 
approximately 


78  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

and  V2  =  —  ql^irca  -f  ql^jrc2d  =  —  Vl 

from  which      Vl—  Vz  —  2  Vl  =  g/27rc-(i/a  —  \J2d)  (42) 

The  capacity  of  the  system  AB  is 

•;;\          s-eKri-rj-nrcKi/a-iitoi)          (43) 

and  the  energy  of  its  field  is 

W=  fl4Trc.(ila-il2d)=4TrcV?l(lla  -  iJ2d)    (44) 

When  the  conducting  sheet  is  placed  coincident  with  the  zero 
equipotential,  the  capacity  of  the  dielectric  between  A  and  this 
surface  is 

s1-e/rl  =  2S  (45) 

The  energy  of  the  dielectric  is 


The  force  tending  to  increase  the  distance  between  the  charge 
at  A  and  the  plate  can  also  be  found  from  (44)  or  (46)  by  the 
method  of  §  55,  I.  Thus 


=—  dWjd(2d)  =- 
dWJdd=  +  dWjd(2d)  =  -  -  2    ™> 


in  agreement  with  (41),  the  first  differentiations  being  performed 
with  the  charges  constant,  and  the  second  with  the  potentials 
constant. 

16.  The  Electric  Field  Surrounding  Two  Concentrated  Charges 
of  the  Same  Sign  in  the  Ratio  of  4  :  1,  and  its  Derivatives.  Max- 
well's diagram  with  twenty  tubes  emanating  from  one  of  the 
charges  (A)  and  five  from  the  other  (j5)  is  given  in  Fig.  23  (from 
Maxwell's  Treatise,  §  118).  One  equipotential  surface,  indicated 
by  the  dotted  line,  consists  of  two  lobes  meeting  at  the  point  P. 
At  P,  which  is  distant  from  A  two  thirds  of  the  distance  AB, 
the  intensity  vanishes.  Within  this  surface,  each  charge  is  sur- 
rounded by  a  separate  system  of  equipotentials,  which  become 


ELECTRIC    FIELDS    AND    CONDENSERS.  79 

more  and  more  nearly  spherical  as  they  become  smaller,  though 
no  one  of  them  is  an  exact  sphere. 

If  two  of  these  surfaces,  one  surrounding  each  point,  are  taken 
to  represent  the  surfaces  of  two  conductors  with  charges  of  the 
same  sign  in  the  ratio  4:1,  the  diagram  will  represent  the  equi- 


Fig.  23. 

potential  surfaces  and  tubes  of  displacement  of  the  field  sur- 
rounding the  conductors,  provided  that  all  the  lines  within  the 
surfaces  are  annulled. 

The  diagram  shows  that  the  force  between  the  two  bodies  will 
be  the  same  as  that  between  the  two  points  A  and  B  with  the 
same  charges.  The  distribution  of  the  tubes  shows  that  this 
force  tends  to  pull  the  bodies  apart. 

If  a  conducting  surface  is  placed  coincident  with  the  two-lobed 
equipotential,  its  electric  surface  density  at  P  will  be  zero  (cf. 

§  57,  I.). 

Outside  the  two-lobed  surface  a  single  system  of  equipoten- 
tials  surrounds  both  charges.  By  making  any  of  these  surfaces 


80  ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

conducting  and  annulling  all  the  tubes  within,  we  obtain  the  field 
surrounding  the  isolated  conductor  with  a  charge  upon  its  sur- 
face equal  to  that  of  A  plus  that  of  B.  The  equipotentials  sur- 
rounding both  A  and  B  approach  the  form  of  spheres  as  their 
distances  from  A  and  B  increase  (cf.  §  57,  I.). 

17.  The  Electric  Field  Surrounding  Two  Concentrated  Charges 
of  Opposite  Signs  in  the  Ratio  4  to  1,  and  its  Derivatives.  Max- 
well's diagram,  with  twenty  tubes  emanating  from  one  charge  at 
A  and  five  terminating  with  the  other  at  By  is  given  in  Fig.  24 
(from  Maxwell's  Treatise,  §  119). 

Here  again  one  of  the  equipotentials,  indicated  by  a  dotted 
line,  has  two  lobes,  an  inner  one  surrounding  the  point  B  and  an 
outer  one  surrounding  both  the  points  A  and  B.  All  the  sur- 
faces in  the  region  between  the  lobes  surround  A  only  and 
become  more  nearly  spherical  as  A  is  approached ;  while  all 
those  in  the  region  within  the  inner  lobe  surround  B  only  and 
become  more  nearly  spherical  as  B  is  approached.  The  equi- 
potentials lying  outside  the  surface  with  two  lobes  become  more 
nearly  spherical  as  their  distances  from  A  and  B  increase. 

One  of  the  surfaces,  that  with  the  potential  zero,  is  a  sphere, 
and  is  indicated  by  the  dotted  circle  Q. 

If  two  of  the  surfaces,  each  surrounding  one  of  the  two  points 
A  and  B,  are  made  conducting,  and  the  fields  within  them  an- 
nulled, the  diagram  gives  the  tubes  and  equipotentials  surround- 
ing these  conductors  when  charged  oppositely  in  the  ratio  4:1. 

The  diagram  indicates  that  the  force  between  two  such 
charged  conductors  is  one  of  attraction,  and  the  same  as  the 
force  between  the  two  charged  points  A  and  B.  The  field  sur- 
rounding the  charge  at  A  or  B  when  the  sphere  Q  is  made  con- 
ducting and  the  field  on  the  other  side  annulled  is  discussed  in 

§23. 

If  we  consider  points  on  the  axis  AB  beyond  the  point  B,  we 
find  that  the  resultant  intensity  diminishes  up  to  the  point/5,  distant 
from  A  twice  the  length  AB,  where  it  vanishes.  It  then  changes 


ELECTRIC    FIELDS    AND    CONDENSERS. 


8l 


sign  and  reaches  a  maximum  at  M,  after  which  it  continually  di- 
minishes. The  distance  of  M  from  A  is  ^4/(^4  —  i)  -  AB 
==  2.70  x  AB  (approximately). 


Fig.  24. 

18.  The  Electric  Field  Surrounding  Three  Points  A,  B,  and  C, 
with  Charges  Proportional  to  15,  —  12,  and  20,  respectively,  so 
Situated  in  a  Straight  Line  that  AB  :  BC  :  AC  : :  9  : 16 :  25,  and  its 

Derivatives.  Maxwell's  diagram  of  the  field  is  given  in  Fig.  25 
(from  Maxwell's  Treatise,  §  121). 

In  this  field  one  of  the  equipotentials,  corresponding  to  the 
potential  I  /4<r,  consists  of  two  spheres  intersecting  at  right  angles, 
with  centers  A  and  C,  and  radii  1 5  and  20,  respectively,  as  indi- 
cated by  dotted  lines  in  the  diagram.  The  point  B  is  at  the 
center  of  the  circle  of  intersection  DD,  the  radius  of  which  is  1 2, 
and  at  all  points  of  which  the  intensity  is  zero. 

If  the  sphere  A  is  made  conducting  and  all  the  lines  within  it 
annulled,  the  diagram  will  represent  the  field  surrounding  the 


82 


ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 


insulated  sphere  A  with  charge  3  upon  its  surface  in  the  pres- 
ence of  a  concentrated  charge  20  at  C.  The  part  of  A  within 
the  spherical  surface  about  C  will  be  negatively  charged  and  the 
rest  positively  charged,  the  electric  surface  density  along  the 
circle  DD  being  zero. 


Fig.  25. 

In  the  same  way,  if  C  is  made  conducting,  the  diagram  repre- 
sents the  field  surrounding  the  conducting  sphere  C  insulated 
with  charge  8  upon  its  surface  in  the  presence  of  the  concen- 
trated charge  15  at  A. 

These  two  fields  are  particular  cases  of  that  discussed  in  §  24. 

If  both  spheres  are  made  conducting,  and  the  lines  within 
annulled,  the  diagram  represents  the  field  surrounding  a  con- 
ducting surface  consisting  of  the  external  segments  of  two  spheres 
intersecting  at  right  angles  in  DD  and  with  charge  23.  This  is 
a  particular  case  of  the  field  discussed  in  §  36. 


ELECTRIC    FIELDS    AND    CONDENSERS.  83 

19.  The  Electric  Field  Terminated  by  Two  Infinite  Parallel 
Straight,  Lines  or  Circular  Conducting  Cylinders,  with  Charges  q 
and  —  q  on  Unit  Length.  Consider  first  two  electrified  straight 
lines,  distant  za  apart,  and  cut  by  a  perpendicular  plane  in  the 
points  Al  and  A2,  Fig.  26.  By  symmetry,  the  distribution  of 
the  lines  of  displacement  is  the  same  in  every  such  plane.  More- 
over, all  the  lines  emanating  from  a  point  Al  pass  to  the  point 
A2  in  the  plane  containing  the  two  points  and  perpendicular  to 
the  two  lines. 


Fig.  26. 

The  potential  at  any  point  P  distant  L^  from  Al  and  L2  from 
A0  is 


=  Vl  +  F2  = 


f 

JL, 


adLjL  -  f  dL]L\ 
JL,  )  £4g) 


dLjL 


This  is  also  the  equation  of  the  section  by  the  plane  of  the 
paper  of  the  equipotential  surface  whose  potential  is  V.  By  giv- 
ing to  V  different  values  the  corresponding  surfaces  may  be 
obtained. 

The  displacement  at  P  is 

D  =  Vector  sum  of  D^(  =  qJ27rLl  directed  from  Al ) 
and  D2(=  q/27rL2  directed  toward  At) 

and  the  intensity  E  is  Djc. 


84  ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

From  (49)  the  equation  of  any  line  of  intensity  can  be  obtained 
by  the  method  of  §  15.  Proceeding  exactly  as  in  that  article, 
we  find  the  equation 

0i  +  02=  C=  constant  (50) 

which  is  evidently  the  equation  of  the  arc  of  a  circle  terminating 
at  Al  and  A2,  and  cutting  perpendicularly  the  line  normal  to 
A^A2  at  its  middle  point.  If  the  line  whose  equation  is  sought 
cuts  the  normal  to  A^A^  at  its  middle  point  O  at  a  distance  x 
from  0,  we  have,  for  this  point,  6l  =  02,  and 

C=20l  =  202=  20=2  cos-1  [*/(«*  +  .r2)*]  (51) 

The  equipotential  surfaces  given  by  (48)  are  circular  cylinders, 
or  their  lines  of  intersection  with  the  plane  A^AJP  circles,  orthog- 
onal to  the  lines  of  intensity.  For  (48)  may  be  written 

fJL2=e-2-^^  =  /i  (52) 

a  constant  for  the  curve,  or  surface,  whose  potential  is  V\  and 
this  is  the  equation  of  a  circle  cutting  the  line  A^A^  and  with  its 
center  C  on  the  line  A^A2  produced. 

The  radius  of  the  circle  whose  potential  is 


-i)  (53) 

the  distance  of  its  center  C  from  Al  is 

A^C-^Rh  (54) 

and  the  distance  of  C  from  A2  is 

A2C=Rjk  (55) 

To  obtain  the  resultant  displacement  D'  at  P  we  must  obtain 
the  vector  sum  of  Dl  and  Z>2,  Fig.  26,  which  will  be  along  R 
normal  to  the  equipotential.  Since  Dl  and  D2  are  directed  along 
L^  and  L2  respectively,  and  since,  by  (49),  DljD2  =  A/A>  the 
triangle  whose  sides  are  Dv  D2,  and  D'  is  similar  to  the  triangle 
A^PA2 ;  so  that 

2a  I L2,  and  Z)f  /D2=  2a/Ll 


ELECTRIC    FIELDS    AND    CONDENSERS. 


85 


whence 


D'  =  2aDl  /L2  =  2aD2  /Zx  =  qa 


(56) 


and  the  resultant  intensity  E  is  equal  to  D'  jc. 

The  force  F  upon  a  length  A  of  either  electrified  line,  con- 
sidered positive  when  tending  to  increase  a,  is 


F=  — 


(57) 


The  plane  diagram  of  the  field,  drawn  by  Maxwell's  method, 
§  14,  is  given  in  Fig.  27  (from  Webster's  Theory  of  Electricity  and 
Magnetism,  §  159).  The  tubes  of  displacement  and  the  equipo- 
tentials  are  mapped  out  by  moving  the  diagram  perpendicularly 
to  its  plane. 


Fig.  27. 

If  for  any  equipotential  surface  the  coincident  surface  of  a  con- 
ductor is  substituted,  the  electric  field  on  the  side  facing  this  sur- 
face will  remain  unaltered.  The  above  field  therefore  includes, 
as  particular  cases,  the  fields  bounded  by 

(i)  An  infinite  straight  line  and  a  parallel  infinite  conducting 
circular  cylinder, 


86     ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

(2)  An  infinite  straight  line  and  a  parallel  infinite  conducting 
plane, 

(3)  An  infinite  conducting  circular  cylinder  and  a  parallel  in- 
finite conducting  plane, 

(4)  Two  parallel  infinite  conducting  circular  cylinders,  internal 
or  external  (either  or  neither  surrounding  the  other),  all  with 
charges  q  and  —  q  upon  unit  length. 

The  fields  of  §§  8—9  are  particular  cases  of  (4)  when  one  of 
the  two  lines  is  removed  to  infinity. 

As  systems  of  practical  importance,  we  shall  discuss  (4)  for 
the  case  in  which  the  two  cylinders  are  external  to  one  another, 
each  of  the  same  given  radius  R,  with  their  axes  at  a  given  dis- 
tance 2d  apart,  and  charged  to  potentials  V  and  —  V,  and  (3), 
which  is  a  particular  case  of  (4). 

To  obtain  the  electric  field  terminated  by  the  two  cylinders,  we 
must  find  the  distance  a  and  the  charge  q  upon  unit  length  of 
the  positive  cylinder. 

From  the  similar  triangles  A^P  and  A2CP  (Fig.  26)  we  have 


a)  =  I?  (58) 

whence 

a  =  (d2-J?y  (59) 

From  (54)  and  (55) 


For  the  cylinder  whose  potential  is  —  J^we  have 

log  h  =  2TTcVjq 
and  therefore 

q  =  27rr  F/log  h  =  27rcF/\og[{d  +  (d*  - 

=   2TTC 


From  (49)  and  (56)  the  field  can  be  determined,  by  making  use 
of  (61),  at  all  points. 

The  capacity  of  a  length  A  of  the  system  is 


ELECTRIC    FIELDS   AND    CONDENSERS.  87 

=  TTcA/log/l  =  7TcA/[og[{d  +  (d2-  ^2)*}/^]      (62) 

and  the  energy  in  the  same  length  is 

W=  \qA  •  2  V-  2-n-cA  F2/log[{rf  +  (rf2  -  *•)»}  JK\ 
=  tfA/*c.loeW  +(<**-  &?}!*] 

If  the  infinite  plane  surface  of  a  conductor  is  placed  coincident 
with  the  surface  of  zero  potential  (the  plane  passing  symmetric- 
ally between  the  conductors)  the  field  on  the  side  facing  the 
conductor  will  remain  unaltered  ;  it  is  simply  half  the  field  just 
considered. 

The  capacity  of  a  length  A  of  the  condenser  formed  by  the 
infinite  plane  and  the  cylinder  with  the  dielectric  is 

5,-^/F-25  (64) 

and  the  energy  is  half  that  contained  in  the  complete  field  sur- 
rounding the  two  cylinders,  or 

',_,....  .  ,    _.  '4         W,  =  \w     _  ^ffff  ;'•;-     (65) 

The  force  F  acting  upon  a  length  of  A  of  either  conductor, 
plane  or  cylindrical,  is  given  by  (57).  It  can  also  be  obtained 
by  differentiating  W^  with  respect  to  d,  or  W  with  respect  to  2d, 
by  the  method  of  §  55,  I.  Thus 

v     ' 

20.  The  Field  of  a  Line  Doublet.  When  20,  is  small  in  com- 
parison with  L^  and  L2,  we  have 


=  (approximately)  g/27rc  •  log  (  I  -f-  2a  cos  0  JR)  (67) 

=  qJ27rc  -  2acos  6  jR(i  —  20,  cos  6  J2R  -\-  •  •  •) 

if  R  is  written  for  L^  and  if  0  denotes  the  angle  between  Zx  and 
the  line  A^AV 

If  now  the  product  q  2a  is  kept  constant  while  a  is  diminished 
indefinitely,  (67)  approaches  the  limit 


88  ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

V=  2aq  cos  6l27rcR  =  J/cos  6 / '  2ircR  (68) 

where  Mis  written  for  2aq.     This  system  is  called  a  line  doublet, 
and  M  is  called  the  moment  of  the  doublet. 


and 


Fig.  28. 

The  radial  and  tangential  displacements  at  a  distance  R  from 
the  doublet,  at  a  point  where  R  makes  an  angle  6  with  the  line 
^  now  infinitely  short,  are 

Dr  =  -  cdVjdR  =  Mcos  6/27rR2  (69) 

Dt=—cd  VjdT=  —  cjR  'dVjdB  =  Msin  dJ2TrR2     (70) 
The  total  displacement  is  equal  to 

D  =  (Dr2  +  Z>,2)*  =  Ml  27rR*  (7  J  ) 

and  makes  an  angle  20  'with  the  line  A^  (the  axis  of  the  doublet). 

The  lines  of  intensity  are  evidently  circles  tangent  to  the  axis 

at  0,  and  the  equipotentials  circles  perpendicular  to  the  axis  at 


ELECTRIC    FIELDS    AND    CONDENSERS. 


89 


O.  The  plane  diagram  of  the  field  is  given  in  Fig.  28  (from 
Webster's  Theory  of  Electricity  and  Magnetism,  §  44),  the  tubes 
of  displacement  and  equipotential  surfaces  being  supposed  gen- 
erated by  moving  the  diagram  perpendicularly  to  its  plane. 

The  method  of  drawing  the  diagram  is  easily  understood  from 
Fig.  29.     Since  there  is  an  infinite  number  of  lines  of  displace- 


Fig.  29. 

ment  within  a  circle  of  any  finite  diameter  a,  only  the  lines  lying 
outside  some  such  arbitrarily  chosen  circle  can  be  drawn.  The 
same  is  true  of  the  equipotential  lines. 

The  flux  through  the  tube  between  the  cylinders  of  unit  depth 
with  diameters  a  and  y  is 


dyfy*  =  M/27T 


Hence  by  giving  II  any  set  of  successive  values  differing  by  a 
constant  the  diameters  {y)  of  the  corresponding  lines  of  displace- 
ment may  be  obtained. 

The  voltage  from  the  circle  of  equal  potential  of  diameter  b  to 
the  circle  of  diameter  x  is 


Hence  by  starting  with  a  circle  of  diameter  b  and  giving  Vb 
—  Vx  any  set  of  successive  values  differing  by  a  constant,  the 
diameters  (x)  of  the  corresponding  equipotential  circles  may  be 
found. 


90  ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

21.  The  Electric  Field  Surrounding  an  Isolated  Conducting- 
Spheroid.  First  it  will  be  shown  that  within  the  region  enclosed 
by  a  homogeneous  material  shell  whose  surfaces  are  similar  and 
similarly  situated  ellipsoids  there  is  no  gravitational  field  of  force. 
Such  a  shell  is  called  an  ellipsoidal  homoeoid. 

Let  a  cone,  Fig.  30,  of  infinitesimal  angle  da>  at  any  point  A" 
in  the  region  cut  from  the  shell  the  volumes  B"  C"  and  D"E"  . 
If  p  denotes  the  density  of  the  shell,  g  the  gravitation  constant, 
L  the  distance  from  A"  of  any  element  of  volume  dr  of  the  shell, 
the  intensity  at  A"  in  the  direction  A"  C"  due  to  the  masses  in 
B"C"  and  D"E"  is 


f*A"C"  s*A"E" 

=  g\         pdrjL^-gi         pttr/L 

J  A"B  "  JA"B  " 


dG 

since  the  attractions  due  to  the  masses  in  J2"C"and  D"  E"  are  in 
opposite  directions.  Since  dr  =  I^dadL,  the  integrals  reduce  to 

dG  =  gpd<*(B"C"  -  D"E") 

and  since  the  plane  N"OC"  intersects  the  ellipsoids  in  two 
similar  and  similarly  placed  ellipses,  the  same  diameter  ON1' 
bisects  B"D"  and  C"E".  Hence  B"  C"  =  C"E"  and 


In  the  same  manner  it  may  be  shown  that  the  intensity  at  A" 
due  to  the  matter  within  any  other  infinitesimal  cone  with  vertex 
at  A"  vanishes.  Hence 


or  the  region  contains  no  gravitational  field. 

When  a  conducting  ellipsoid  is  charged  the  tubes  of  displace- 
ment are  distributed  by  the  tensions  and  pressures  until  they 
touch  the  surface  normally,  or  until  the  surface  becomes  an  equi- 
potential.  The  conducting  substance  may  then  be  considered 
replaced  by  a  dielectric  of  permittivity  c  equal  to  that  of  the  exte- 
rior medium,  for  the  sake  of  applying  the  law  of  inverse  squares, 
§  28,  I.  At  any  point  A  within  this  region  the  intensity  and  dis- 


ELECTRIC    FIELDS    AND    CONDENSERS. 


placement  are  zero.  And  therefore,  since  the  law  of  inverse 
squares  prevails  in  electrostatics  as  in  gravitation,  the  law  of 
variation  of  the  electric  surface  density,  or  outward  displacement 
at  the  surface,  must  be  identical  with  the  law  of  variation  of  the 
thickness  of  the  shell  in  the  above  gravitational  problem.  In 
discussing  the  electric  case  the  shell  must  be  considered  as  of 
infinitesimal  thickness,  since  the  charge  resides  wholly  at  the 
surface  of  the  conducting  ellipsoid. 

The  distribution  of  charge  or  displacement  at  the  surface  of  an 
ellipsoid  of  revolution,  or  spheroid,  only  will  be  determined  here. 
We  proceed  to  find  the  distribution  of  the  charge  by  investigating 
the  law  of  the  variation  of  the  thickness  of  a  spheroidal  homceoid. 


c" 


Let  the  given  spheroid  be  generated  by  the  revolution  of  the 
ellipse  BAB' ,  Fig.  30,  about  the  line  BBf ,  and  the  exterior  sur- 
face of  the  infinitely  thin  homceoid  by  the  revolution  of  the  sim- 
ilar ellipse  B^A^BJ  about  the  same  axis.  We  must  find  the  ratio 
of  /,  thickness  of  the  shell  at  any  point  P,  to  /0,  the  thickness  at 
B.  Since  the  shell  is  infinitely  thin,  t  is  to  be  measured  in  the 
direction  of  the  normal  PJPC  to  the  spheroid  at  P. 

From  the  similar  triangles  />/>/>  and  PCD  we  have 

*  =  PAS/PC-  ay(y,  -  y)\b(<?  -  *V)»  =  y(y,  -  y)  OR\P 
By  the  equations 


92  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

y?  -f  =  VK  •  W  -*?}-  v-l#  •  ('f  -  **), 

b\la\  =  &/a  (since  the  spheroids  are  similar), 
and 

x  =  xv     y  —  y^     (in  the  limit), 
we  have 

-JO  =  b(b,  -6)  =  bt, 


and  the  equation  for  /  becomes 

t/t,  =  bjPC  =  al(<?  -  eW)>  =  OR  I  'b  =  DID,  =  <r/<r0 


or 
D=cr  =  <rQa/(a2  -  e2*2)*  =  DQal(a2  —  *2e2)*  =  DJb  =  crQ  jb  (72) 

where  crQ  and  cr  denote  the  electric  surface  densities  at  B  and  at 
points  of  the  spheroid  distant  x  from  the  axis  BB' ',  and  J90  and 
D  the  corresponding  outward  displacements. 

Thus  the  surface  density  increases  in  passing  from  B  to  A,  at 
which  point  it  has  the  value 

er  =  <TQa!:b  (73) 

To  obtain  the  total  charge  q  upon  the  spheroid,  the  charge 
<rdS  upon  the  elementary  zone  cut  out  by  PP2  as  the  ellipse  re- 
volves about  OB  must  be  found  and  integrated  over  the  whole 
surface.  Thus 


<rdS  =  <T27rxPP2  =  bcr^  JPC-  2-jrxdx-  PC\y  =  -  2-rrcr^lb  •  dy 
and 

q  =    I  crdS  =  —  27rtf2<70/£  I      dy  =  4ira?<r0  (74) 

J  J+b 

The  potential  of  the  spheroid  when  the  charge  is  q  may  be  ob- 
tained by  rinding  the  potential  at  the  center  0,  since  the  potential 
is  uniform  over  and  within  the  spheroid. 

The  potential  at  0  due  to  the  charge  upon  the  zone  dS  is 


ELECTRIC    FIELDS   AND    CONDENSERS.  93 

and  the  total  potential  is 

*  (75) 


The  capacity  of  the  isolated  spheroid,  or  the  permittance  of  the 
dielectric  bounded  by  the  spheroid  and  an  infinitely  remote  sur- 
rounding surface  (§  58,  I.),  is 

(76) 


The  isolated  sphere.  If  b  =  a,  the  equations  reduce  to  those 
already  developed  (§  2)  for  an  isolated  sphere. 

An  infinitely  thin  circular  plate.  If  b  —  o,  the  spheroid  re- 
duces to  an  infinitely  thin  circular  conducting  plate  of  radius  a, 
and  (72),  (74),  (75),  and  (76)  become 

D  (or  *)  =  aD9(or  a  J  /(*-**)*  (77) 

since  e  =  [(a2  —  &2)/a2~\*  =  I, 

q  =  4^2°"0  =  4w^A 

F=  7rtf<70/2r  (79) 

S  =  8tfr  (80) 

Near  the  edges  the  displacement  is  very  great,  D  becoming 
infinite  when  x  =  a.  But  the  total  charge  is  finite  since  an  edge 
has  no  area. 

The  ratio  of  the  capacity  of  a  thin  circular  plate  to  that  of  a 
sphere  of  the  same  radius  is 

8ac/4.7rac=  2/7T  =  1/1.571 

a  relation  established  experimentally  by  Cavendish  long  before 
the  development  of  the  theory. 

If  two  thin  circular  plates  of  the  same  radius  a  are  placed  par- 
allel to  one  another  with  the  distance  d  between  them  so  small 
that  the  field  is  sensibly  uniform  between  them,  and  relatively 


94  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

weak  outside,  the  capacity  of  the  system  will  be  very  approxi- 
mately 


The  ratio  of  the  capacity  of  a  single  isolated  plate  to  this 
capacity  is 


The  energy  of  the  field  connected  with  two  such  plates  very 
remote  from  one  another  and  with  charges  q  and  —  q,  respec- 
tively, is 


The  energy  of  the  field  when  the  two  plates  are  parallel  and 
separated  by  the  very  small  distance  d  is 


ira'c 


Hence  the  work  which  would  be  done  by  the  electrical  forces 
in  drawing  the  two  plates  into  the  latter  configuration  from  the 
former  is 

f(  I  fSac  —  dJ2ira2c)  =  <f  j  2ac  -(1/4  —  djira) 

It  may  be  shown  that  the  equipotential  surfaces  surrounding 
the  isolated  plates  are  the  confocal  spheroids  with  the  edge  of 
the  plate  as  focal  line,  and  that  the  lines  of  displacement  are  the 
corresponding  confocal  hyperbolas. 

If  any  one  of  the  spheroidal  equipotentials  is  made  conducting, 
and  the  lines  within  annulled,  the  remainder  of  the  field  just 
described  will  be  the  field  surrounding  this  spheroid. 

22.  The  Average  Value  of  the  Potential  over  a  Spherical  Sur- 
face in  any  electric  field  whose  charges  are  situated  wholly  out- 
side of  or  upon  the  sphere  and  whose  dielectric  is  homogeneous 
and  isotropic  within  the  sphere  is  equal  to  the  potential  at  its 
center.  To  prove  this,  consider  first  the  spherical  surface  5  of 
radius  R  in  the  radial  field  from  a  concentrated  charge  q  distant 
x  from  the  center  of  the  sphere,  q  being  the  only  charge  in  the 


ELECTRIC    FIELDS    AND    CONDENSERS. 


95 


field.      Let  the  charge  be  at  A  and  the  center  of  the   sphere  at 
C,  Fig.  31. 

The  area  of  an  elementary  zone  of  the  sphere  included  be- 
tween two  planes  perpendicular  to  A  C,  at  distances  y  and  y  +  dy 


The  potential  of  this  zone  is 


Vy  = 


The  integral  of  the  potential  over  the  zone  is  therefore 
VdS  =  qRdyJ2c  (R2  -  #  + 


Fig.  31. 


To  obtain  the  average  value,  V,  of  the  potential  over  the  sur- 
face of  the  whole  sphere,  the  integral  of  this  expression  must  be 
taken  over  the  sphere,  and  the  result  divided  by  the  area  of  the 
sphere.  Thus 


V 


=  i/4^J 


VdS  = 


-f  R 


dyj(R2 


+ 


(Si) 


which  establishes  the  proposition  for  a  single  concentrated  charge. 
If,  instead  of  a  single  concentrated  charge,  there  is  any  other 
electric  distribution,  subject  to  the  limitations  above  mentioned, 
we  have  by  the  principle  of  superposition 


96  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

which  establishes  the  proposition  for  a  homogeneous  dielectric, 
or  such  a  dielectric  and  conductors,  filling  all  space.  That  the 
proposition  is  perfectly  general  will  appear  from  Chapter  IV.,  (17). 
As  an  example,  the  average  potential  of  the  insulated  sphere 
of  §  24,  which  is  the  same  as  the  potential  of  any  point  of  the 
sphere,  since  it  is  a  conducting  surface,  is 

V=  qqircx  + 


since  ^,  the  sum  of  the  (induced)  charges  on  the  sphere,  is  zero. 
If,  in  addition  to  the  induced  charges,  the  sphere  possesses  a 
charge  q,  its  potential  is 

V=  q^qtrcx  +  qJ4TrcR 

The  same  results  are  obtained  by  another  method  in  §  24. 
They  also  follow  immediately  from  (15)  and  §  28,  I. 

23.  The  Electric  Field  Surrounding  a  Concentrated  Charge  in 
the  Presence  of  a  Spherical  Surface  at  Zero  Potential.  Consider 
first  the  case  in  which  the  given  charge,  qv  is  at  Al  external  to 
the  sphere,  S,  Fig.  26.  Let  R  denote  the  radius  of  6"  and  x  the 
distance  between  its  center  and  the  charge  qr 

If  a  charge  q2  and  its  position  within  the  surface  .S  can  be 
found  such  that  in  the  field  surrounding  ql  and  qz  S  is  a  surface 
of  zero  potential  when  there  are  no  other  charges  in  the  field, 
then,  by  §  48,  I.,  the  portion  of  the  field  outside  5  will  be  the 
only  field  satisfying  the  given  conditions,  and  q^  and  q2  will  be 
the  electric  images  of  one  another  in  the  surface  5  if  it  is  made 
conducting. 

If  a  charge  q2  is  placed  at  A2,  the  potential  at  a  point  P  of  the 
sphere  will  be 

V=  Vl+V,=  I/4W  fo/Z,  +  gJLJ  =  1/4^^  •  (9l  +  hg,} 

where  h  =  L^L^  has  the  same  significance  as  in  §  19. 

If  q2  is  so  chosen  that 

+        =  o 


ELECTRIC    FIELDS    AND    CONDENSERS.  97 

V  '  =  o  for  every  point  upon  the  sphere.  Hence  the  portion  out- 
side 5  of  the  field  surrounding  the  charge  ql  at  Al  and  q2  at  A2, 
where 

A-  -ft/*  (82) 

is  the  field  required. 

Al  and  A2  are  called  inverse  points,  or  geometrical  images  of  one 
another,  in  the  sphere  (§  30). 

The  part  of  the  field  within  .S"  is  the  field  surrounding  a  con- 
centrated charge  q2  in  the  interior  of  a  spherical  surface  at  zero 
potential,  the  charge  being  distant  A2C2  =  xf  from  the  center  of 
the  sphere. 

The  resultant  displacement,  D,  at  any  point  of  either  of  the 
required  fields  (given  charge  inside  or  outside  the  sphere)  is  thus 
the  vector  sum  of  Dl  and  D2,  the  radial  displacements  which 
would  accompany  the  charges  ql  and  q2  separately.  We  shall 
find  the  displacement  only  at  the  surface  of  the  sphere,  to  which, 
an  equipotential,  it  is  everywhere  normal. 

When  gl  is  positive,  and  q2  therefore  negative,  Dv  D2,  and  D 
are  in  the  directions  A^P,  PA2,  and  PC,  respectively,  their  direc- 
tions being  reversed  when  ql  and  qz  change  signs.  Moreover,  in 
magnitude, 


A/A  =  fe/4TA2)/(?,/4TA2)  =  k  =  *IR  =  Rl*>  =  A/A 

from  the  geometry  of  §  19.     Therefore  the  triangle  with  sides 
Dv  D2  and  D  is  similar  to  the  triangle  AfAv  and 


in  magnitude. 

If  D  is  reckoned  positive  along  the  outward  normal  to  S,  we 
have,  therefore, 


D  =  -  MhqjAnL*  =  -  9lR(»  -  i)/4"A3 

•  (x  -  R%*  +  R)jR  (* 


which  is  equal  to  the  electric  surface  density  on  the  outside  of 
the  sphere  at  P,  when  the  sphere  is  the  surface  of  a  conductor  at 


98  ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

zero  potential  in  the  presence  of  the  charge  q^  at  Ar     The  total 
charge  upon  the  sphere  is  evidently  qr 

If  D  is  reckoned  positive  along  the  inward  normal  to  S,  we 
have 

D  =  q£(1?  -  I)/47TA3  =  qtf  ~  &)\4*RL?  (84) 


which  is  equal  to  the  electric  surface  density  on  the  inside  of  the 
sphere  at  P  when  the  sphere  is  the  inner  surface  of  a  conductor 
at  zero  potential  surrounding  a  charge  q2  =  —  qljh  at  A2.  Since 
when  the  internal  field  is  sought,  qv  x*  ',  and  L2  are  the  given 
quantities  instead  of  qv  x,  and  Lv  (84)  must  be  transformed  by 
substituting  for  qv  x,  and  Lv  their  equals  —  qjt  =  —q^Rjx',  R2/x', 
and  hL2  =  RLJx*  ',  respectively.  On  making  these  substitu- 
tions, we  have 


D  =  - 


The  total  charge  on  the  inner  surface  of  the  sphere  is  — 
The  force  between  either  charged  body  and  the  sphere  is 


Maxwell's  plane  diagram  of  the  field,  for  the  case  in  which 
h=  —  ^i/^2  =  2°/5  is  given  in  Fig.  24,  the  sphere,  of  radius 
R  =  ABfh  =  \AB,  being  indicated  by  the  dotted  circle  Q. 

If  the  radius  R,  in  what  precedes,  is  made  to  approach  infinity, 
while  (x  —  R)  or  (R  —  xr)  is  kept  constant,  we  have,  in  the  limit, 
a  concentrated  charge  in  the  presence  of  an  infinite  plane  surface 
at  zero  potential,  and  the  above  equations  reduce  to  the  equa- 
tions of  §  15. 

24.  Sphere  at  Any  Potential,  or  with  Any  Charge,  in  the  Pres- 
ence of  a  Concentrated  Charge.  The  field  external  to  the  sphere 
in  §  23  is  a  particular  case  of  the  field  surrounding  a  concen- 
trated charge  and  a  spherical  equipotential  surface  (as  that  of  a 


ELECTRIC    FIELDS    AND    CONDENSERS.  99 

spherical  conductor),  the  potential,  V,  of  the  surface,  or  the  out- 
ward flux  across  it  (or  charge  upon  it,  if  a  conducting  surface), 
<?,  being  given. 

I  .  If  the  potential  V  is  given,  the  required  field  is  found  by 
superposing  upon  the  external  field  of  §  23  the  radial  field  from 
a  charge  qz  =  ^nrcRV  at  the  center  of  the  sphere.  For  the  po- 
tential at  every  point  of  the  sphere  due  to  the  fields  from  ql  and 
q.2  is  zero,  and  the  potential  due  to  the  radial  field  from  qz  is  the 
same  at  every  point  of  the  sphere  and  equal  to 

qJqircR  =  47rcRV/4.7rcR  =  V 

so  that  the  resultant  potential  is  the  same  all  over  the  sphere  and 
equal  to  V.  The  field  outside  the  sphere  is  therefore  determined 
(§  48,  I.)  and  is  identical  with  the  field  surrounding  the  charge 
qi  and  a  spherical  conductor  of  radius  R  and  at  potential  V. 
The  total  charge  upon  the  conductor  is  q  =  q2  +  qy 

2.  If  the  outward  flux  q  (or  total  charge,  if  the  surface  is  that 
of  a  conductor)  is  given,  the  required  field  is  found  by  superpos- 
ing upon  the  external  field  of  §  23  the  radial  field  from  a  charge 
gz  =  q  —  q2  placed  at  the  center  of  the  sphere.  For,  as  in  (i), 
the  surface  will  remain  equipotential,  and  the  flux  across  it  (or  the 
charge  upon  it)  will  be  (q  —  q^)  -\-  qc>  =  q.  The  position  of  the 
equipotential  surface  being  given  together  with  the  flux  across  it 
and  the  outside  charges,  the  external  field  is  determined  (§  48,  I.) 
and  is  identical  with  the  field  surrounding  a  conducting  sphere 
of  radius  R  and  charge  q.  The  potential  of  the  sphere  is 

V= 


The  outward  displacement  at  the  surface  of  the  sphere  (or  the 
electric  surface  density  if  the  equipotential  sphere  is  the  surface 
of  a  conductor)  is 

D  =  Dz  +  D'(  =  D  of  preceding  article)  =  fJfirlP 

\ 

(2)      S  7) 
=  (q  +  giRlx)lvr&  -  qtf  -  PP)lvrRL*  (3)  . 


100          ELEMENTS   OF    ELECTROMAGNETIC   THEORY.         . 

Maxwell's  plane  diagram  of  this  field  is  given  for  two  particu- 
lar cases  in  Fig.  25,  and  is  discussed  in  §  18. 

The  force  between  the  body  with  the  charge  q^  and  the  spher- 
ical conductor  is  equal  to  the  force  between  this  body  and  those 
with  the  charges  q.z  and  q^  Thus 


F  =  ?i?2/4^(2tf  )2  +  q  ,  . 

-  R2)2  +  q,(q  +  q,R  /  x)  /  'fire*        ^  *  \  (88) 


-  R2}2  +  q,  VRjx2  (2) 

If  the  sphere  is  insulated  without  algebraic  charge,  q  =  q2  -f  q^ 
o,  and  £3  =  —  qz  =  q^Rjx.     In  this  case  (87)  and  (88)  become 


D  =  qJvrR  •  [i  I*  -  (*»  -  R*)ILft  (89) 

and 

F=  -  q?Rxl4irc(**  -  R2)2  +  q2Rl4-jrc^  (90) 

If  the  given  charge  is  internal,  the  internal  field  is  exactly  the 
same  as  that  in  §  23,  and  the  external  field  is  the  radial  field 
from  the  charge  q  at  the  center  of  the  sphere. 

If  a  conducting  sphere  is  in  the  presence  of  any  number  of 
fixed  charges,  internal  and  external,  the  electric  images  and  the 
electric  field  can  be  got  at  once  from  what  precedes  by  the  prin- 
ciple of  superposition. 

25.  A  Conducting  Sphere  in  a  Uniform  Field.  If  q  =  o,  §  24, 
and  if  q^  is  kept  constant  and  x  made  to  increase  without  limit, 
Dl  approaches  a  uniform  direction,  parallel  to  xy  and  a  uniform 
magnitude,  at  all  points  within  and  near  the  sphere.  But  this 
uniform  magnitude  is  zero.'  If,  however,  as  x  increases,  ql  is 
made  to  increase  at  such  a  rate  that  qjqirs?  remains  constant  and 
equal  to  DQy  then,  when  x  =  infinity, 


and  we  have  an  insulated  spherical  conductor  in  a  field  whose 
displacement  is  uniform  (except  for  the  disturbances  due  to  the 
presence  of  the  sphere)  and  equal  to  DQ. 


ELECTRIC   FIELDS   AND    CONDENSERS., .,,,,    ,JQ I  . 

We  can  obtain  the  resultant  displacement  D  at  the  surface  of 
the  sphere  by  substituting  in  (89)  for  q^  its  value  ^irx^D^  and  for 
Zt  its  value  (^  +  R2  +2Rx  cos  0)*,  where  0  is  the  angle  PCE, 
Fig.  26,  reducing,  and  passing  to  the  limit  x  =  infinity.  Thus 

D  =  DJR  -  {  [x(x2  +  R2  +  2Rx  cos  0)* 

2Rx  cos  0)1} 

cos  e\x)  4-  . . . 

+  2^  COS  0/4T+  •  .  •)]  (90 

=  (3^  cos  6  -f  terms  containing  powers  of  x  in  de- 
nominators)^ I  +  terms  containing  powers  of  x 
in  denominators) 

=  3^o  cos  e 

in  the  limit,  the  displacement  for  a  given  value  of  the  angle  0 
being  thus  independent  of  the  radius  R  of  the  sphere. 


Fig.  32. 

At  the  poles  of  the  sphere,  where  6  =  o°  and  180°,  D  =  + 
3-Z?0  and  —  3^0,  respectively ;  while  its  value  at  the  equator, 
where  9  =  90°,  is  zero. 

The  displacement  at  a  point  of  the  infinite  plane  passing 
through  the  equator  and  distant  L  from  the  center  of  the  sphere 
is  (§  26)  the  sum  of  DQ  and  the  displacement  —  M  sin  go°/47rL5 
due  to  the  doublet  of  moment  M=  4.TrRzDQ  at  the 


102    ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

center  of  the  sphere  with  its  axis  in  the  direction  of  the  displace- 
ment D^.  Thus  at  such  a  point 

'';-  D=D^~R*!D)  V          (92) 

The  plane  diagram  of  the  field  drawn  by  superposing  accord- 
ing to  the  method  of  Maxwell,  §  14,  the  diagram  of  the  doublet 
(§  27)  of  moment  M =  47r/?3Z>0  and  that  of  the  uniform  field  of 
displacement  DQ  is  given  in  Fig.  32  (from  Webster's  Theory  of 
Electricity  and  Magnetism,  §  194).  The  diagram  of  the  field 
surrounding  the  conducting  sphere  is  obtained  from  this  figure 
by  annulling  the  lines  within  the  circle  (see  Fig.  62). 

From  symmetry,  it  is  evident  that  there  is  no  resultant  force 
upon  the  conductor. 

A  Hemispherical  Boss  upon  an  Infinite  Plane.  All  the  lines  of 
intensity  in  the  above  field  (except  of  course  those  terminating 
upon  the  sphere)  cut  the  infinite  plane  (an  equipotential,  with  the 
potential  zero)  passing  through  the  equator  normally.  Hence, 
if  this  surface  is  made  conducting,  the  fields  on  each  side  will  re- 
main unaltered  and  each  will  be  the  field  proceeding  from  (or  to) 
an  infinite  plane  conductor  with  a  hemispherical  boss  upon  it. 
The  surface  density  upon  the  boss  is  given  by  (91),  and  that  upon 
the  rest  of  the  surface  by  (92),  or  by  this  expression  with  the 
opposite  sign,  according  to  the  half  of  the  original  field  con- 
sidered. 

26.  The  Field  of  an  Electrical  Point  Doublet.     Method  I.     In 

the  problem  of  §  25,  as  x  approaches  the  limit  infinity,  q^  =  —  q2 
=  qYRjx  =  ^.D^irRx  also  approaches  the  limit  infinity,  while  the 
distance  Af  =  R2jx  approaches  the  limit  zero  at  the  same  rate. 
The  product  qz-  A2C  therefore  remains  finite  and  constantly  equal 
to  47rft3Z)0.  Such  a  system,  consisting  of  two  equal  and  opposite 
very  great  charges  at  a  very  small  distance  apart,  when  indefi- 
nitely near  its  limit,  is  called  an  electric  point  doublet,  and  the  finite 
product  of  the  positive  charge  by  the  distance  between  them  is 
called  the  moment  of  the  doublet.  The  straight  line  directed  from 
the  negative  to  the  positive  charge  is  called  the  axis  of  the  doub- 


ELECTRIC    FIELDS    AND    CONDENSERS. 


103 


let.      In    the    case    considered,    the    axis    of  the  doublet  is  the 
line  A2  C,  and  the  moment  is 

M-firKD,  (93) 

The  total  displacement  D  in  the  field  of  §  25  is  therefore  the 
vector  sum  of  the  displacement  Dd  due  to  the  field  connected 
with  the  doublet  and  the  uniform  displacement  DQ  directed  paral- 
lel to  the  axis  of  the  doublet. 


>Axis 


Fig.  33. 

In  many  cases  it  is  necessary  to  know  the  field  of  a  doublet 
alone,  that  is,  to  know  the  displacement  Dd.  This  quantity  can 
be  easily  found  by  taking  the  vector  difference  of  D  and  Z>0, 
§25. 

From  the  figure  (Fig.  33)  it  is  evident  that  the  radial  com- 
ponent of  Dd  at  P,  a  point  whose  coordinates  are  R  and  6,  is 

D  =    /)  cos  6  —  D  cos  d  =  2D  cos  0  =  2Mcos 


MCOS  0/2TTR3 


measured  in  the  direction  of  increase  of  R. 
The  tangential  component  is 

=  Msin 


Dt  = 


sn 


measured  in  the  direction  of  increase  of  6. 

The  total  displacement  Dd  due  to  the  doublet  is 

Dd  =  (D?  +  Dft  =  M/47rR*  -  (3  cos2  0  +  i) 


(94) 


(95) 


(96) 


104          ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 
The  horizontal  and  vertical  components  are 


and 


\Q      N         (97) 
3  cos2  6  —  i) 

Z>r  =  D  sin  0  =  3Z>0  sin  6  cos  0  =  ^M/47rR3  •  sin  0  cos  0  (98) 
The  angle  made  by  Dd  with  the  axis  of  the  doublet  is 

6'  =  sin-lDJ£>d  =  sin-1  [3  sin  0  cos  Of  (3  cos2  6  - 1)]    (99) 

27.  The  Field  of  an  Electrical  Point  Doublet.     Method  II.     In 
§  26  the  field  of  a  doublet  was  obtained  by  using  the  results  of 


Fig.  34. 

§  25.  The  field  may  be  found  independently  as  follows.  We 
shall  first  obtain  an  approximate  solution  for  the  case  of  two 
charges  q  and  —  q  separated  by  a  distance  L,  short  in  compari- 
son with  OP=R,  Fig.  34,  the  product  qL  being  equal  to  M. 
Then  if  M  is  kept  constant  while  L  is  increased  indefinitely,  the 
system  becomes  a  doublet  and  the  solution,  in  the  limit,  exact. 
The  potential  at  P  is 


ELECTRIC    FIELDS    AND    CONDENSERS.  105 

But  L2  —  L^  =  L  cos  9  approximately,  and  L^L^  =  R2  approxi- 
mately, these  relations  approaching  exactness  indefinitely  as  L 
approaches  zero.  Hence,  in  the  limit;  for  a  doublet, 


V=  qL  cos  0/4-TrcR2  =  J/cos  0  /  ^ircR2  (100) 

The  radial  component  of  the  displacement  is 

Dr  =  cEr  =  -  cdV\dR  =  Mcos  6/27rR3  (94) 

and  the  tangential  component  is 
Dt  =  cEt=-  cdVjdT=  -  cdV\de  •  dQ\dT  =  Msin  (9/47r^3  (95) 

since  dT=  Rdd,  or  dOldT=  i/R. 

From  the  above  equations  the  horizontal  and  vertical  compo- 
nents, Dh  and  Dv,  and  the  angle  0r  made  by  Dd  with  the  axis  of 
the  doublet,  can  be  readily  found.  These  quantities  are  given  in 
the  preceding  article. 


Fig.  35. 

The  equation  of  a  line  of  intensity  or  displacement,  L,  is  easily 
found  from  (94)  and  (95)  with  the  assistance  of  Fig.  35.  For 
we  have  evidently 

DrjDt  =  2  cos  0/sin  0  =  dRjRdO 
from  which  we  obtain 

dRjR  =  2  cos  6d0/sm  0 
The  integral  of  this  equation  is 

(101) 


io6 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY, 


which  is  the  equation  sought,  C  being  a  constant  for  a  given  line 
and  equal  to  the  distance  from  the  doublet  at  which  the  line  cuts 
the  plane  perpendicular  to  the  doublet's  axis  through  its  center. 
When  C  is  given  for  any  line,  the  line  can  be  drawn  directly 
by  means  of  (101),  or  it  can  be  drawn  by  the  process  indicated  in 
Fig.  36.  Draw  a  line  OA  making  an  angle  6  with  OX,  which  passes 
through  the  axis  of  the  doublet,  and  cutting  in  the  point  A  a 
circle  with  radius  C  and  0  as  center.  Then  let  fall  a  perpen- 
dicular AB  on  the  line  OYy  perpendicular  to  OX  through  the 
doublet,  and  a  second  perpendicular  from  B  on  OA  cutting  it 
in  P.  P  is  a  point  on  the  line  required.  For  OB  =  C  sin  6,  and 
OP  =  OB  sin  0  =  C  sin2  0.  See  J.  Buchanan,  Nature,  Vol.  21, 
1880,  p.  370. 


If  the  figure  is  rotated  about  the  axis  OX,  the  flux  through 
the  tube  enclosed  between  the  surfaces  for  which  C  =  C  and 
C  =  a  will  be 


11=    C 

Ja 


dCjC2 
=  M/27r-(i/a-  i/C) 


Hence  by  giving  to  II  a  series  of  successive  values  differing 
by  a  constant  quantity  and  starting  with  a  curve  for  which  C  has 
an  arbitrary  value  a.  the  value  of  C  can  be  determined,  and  the 
lines  of  displacement  drawn  to  correspond  to  tubes  of  equal 
strength,  for  as  much  of  the  field  as  desired. 


ELECTRIC    FIELDS    AND    CONDENSERS.  IO/ 

The  equipotential  surfaces  corresponding  to  successive  equal 
potential  differences  can  be  drawn  by  means  of  (100),  which  may 
be  written 


and  which  becomes,  for  points  along  OX, 


Fig.  37. 

The  plane  diagram  of  the  lines  of  displacement  is  given  in 
Fig.  37  from  Webster's  Theory  of  Electricity  and  Magnetism. 
§  122)  and  the  diagram  of  both  lines  of  displacement  and  equipo- 
tentials  for  the  part  of  the  field  lying  outside  a  sphere  with  the 
doublet  at  its  center  in  Fig.  67. 


io8 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


28.  The  Electric  Field  Bounded  by  a  Concentrated  Charge  q 
and  Two  Infinite  Planes  at  zero  potential  meeting  at  an  angle 
0  =  Tr/n,  where  n  is  an  integer.  We  shall  find  the  image  sys- 
tem of  the  charge  q  in  the  planes.  From  this  system  and  the 
given  charge  q  the  electric  field  at  all  points  can  then  be  deter- 
mined by  methods  already  discussed.  Let  P  denote  the  position 
of  the  given  charge,  and  let  its  distances  from  the  two  planes  be 
denoted  by  a  and  b  respectively. 

Case  I.  When  n  =  1 ,  6  =  TT,  the  two  planes  are  coincident, 
a  =  by  and  the  image  of  q  is  a  concentrated  charge  —  q  distant  from 


-q, 

X 

x 

j  \ 

I 

V         \ 

I 

!    \ 

Y      1 

T  — 
I  . 

1      i 

\ 
\. 

»;/' 

\ 

i  / 

\ 

L 

-^' 

ta 

•  X^-Q 

Fig.  38. 


Fig.  39. 


q  2a  =  2b  on  the  other  side  of  the  plane  on  the  line  through 
P  perpendicular  to  the  plane.  This  field  is  fully  discussed  in  §  15. 

Case  II.  When  n  =  2,  0  =  ir/2,  and  the  image  system  con- 
sists of  the  charges  -f  q  and  —  q  situated  as  shown  in  Fig.  38  at 
the  corners  of  a  rectangle  with  sides  2a  and  2b  and  on  the 
circle  of  radius  (#2  -f  £2)*  with  center  at  the  intersection  of  the 
planes. 

Case  III.  When  n  =  3,  6  =  ?r/3,  and  the  image  system  con- 
sists of  charges  situated  as  shown  in  Fig.  39. 


ELECTRIC    FIELDS    AND    CONDENSERS. 


109 


Case  IV.  The  images  when  n  is  any  other  integer  may  be 
obtained  in  the  same  manner.  They  consist  in  every  case  of 
concentrated  charges  +  q  and  —  q  distant  a  and  b  from  the 
planes  and  situated  symmetrically  on  the  circle  through  the 
given  charge  at  P  and  with  center  at  the  intersection  of  the 
planes.  The  charges,  in  going  around  the  circle,  are  alternately 
4-  and  — . 

Case  V.  When  a  and  b  are  kept  constant  and  n  is  made  infi- 
nite, we  have  a  concentrated  charge  between  two  parallel  planes, 

I 


Fig.  40. 

Fig.  40.  In  this  case  the  arc  of  the  circle  through  P  is  a  straight 
line,  and  the  images  are  located  along  this  line  at  intervals  of  2a 
and  2b. 

In  all  the  above  fields  the  total  charge  upon  the  two  planes 
is-?. 

29.  The  Electric  Field  Surrounding  a  Conducting  Surface  Formed 
of  the  External  Segments  of  Two  Spheres  Intersecting  at  Right 
Angles  and  Maintained  at  Any  Potential  V.  The  field  will  be 
found  by  the  method  of  images. 


110          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

Let  a  and  b  denote  the  radii  of  the  two  spheres  with  centers  at 
A  and  C,  respectively,  Fig.  25,  distant  d  apart.  Since  the  spheres 
intersect  at  right  angles,  we  have  from  the  figure 

BD  =  abj(a*  +  &2)*  =  abjd 
and  ABAC=P  =  ABd 

CB  CA  =  a2  ==  CB  d 

Hence  B  is  the  geometrical  image  of  A  in  the  sphere  C  and 
the  image  of  C  in  the  sphere  A.  If  therefore  a  charge  ql 
—  4.7rca  V  is  placed  at  A,  a  charge  q^  =  Aprcb  Fat  (7,  and  a  charge 
<72  =  —  4?rr  j§Z)  =  —  qirc  abjd  =  —  ql  b[d  =  —  qz  ajd  at  B,  the 
(uniform)  potential  of  the  sphere  A  due  to  the  three  charges  will 
be  qjqjrca  =  V,  since  the  potential  over  A  due  to  q2  and  q^  is 
zero  (§  23);  and  the  (uniform)  potential  of  the  sphere  C  will  be 
qjqircb  =  V,  in  the  same  way.  Hence  both  spheres  are  at 
potential  V  in  the  presence  of  the  three  charges.  Hence  the 
portion  outside  the  surface  of  the  field  connected  with  these 
charges  is  the  field  sought. 

The  plane  diagram  of  the  field  connected  with  the  charges 
at  A,  B,  and  C,  when  bja  =  20/15  and  ql  =  +  15,  q^  =  -f  20. 
q2  =  —  12,  is  given  in  Fig.  25,  in  which  the  spheres  are  shown 
in  dotted  lines.  At  D,  the  circle  of  intersection,  the  displace- 
ment is  zero,  in  accordance  with  §  57,  I.  The  field  is  further 
discussed  in  §18  above. 

The  charges  qv  qv  q^  will  be  found  in  another  manner  in  §  36. 

The  total  charge  upon  the  conductor  is  equal  to  the  algebraic 
sum  of  the  charges  at  A,  B,  and  C.  Thus 

q  =  ^  +  ?2  +  ?3  =  A^cV\_a  +  b-  abj(a*  +  J2)*]      (102) 
The  capacity  of  the  conductor  is 

S=qjV=  ^rc\a  +  b-  ab  /(a*  +  J2)*]  (103) 

The  charge  upon  the  spherical  segment  A,  or  the  flux  across 
this  segment  from  the  images  at  A,  B,  and  C,  is  easily  seen  to  be 


27TC 


ELECTRIC    FIELDS    AND    CONDENSERS.  1 1 1 

The  charge  upon  the  segment  C,  found  by  interchanging  a  and 
b  in  (104)  is  2TrcV\a  +  b  +  (a*  -  P  -  ab)jd]  (105) 

If  one  of  the  spheres  is  made  infinitely  greater  than  the  other, 
the  problem  reduces  to  that  of  a  hemispherical  boss  upon  an 
infinite  plane,  §  25. 

30.  Geometrical  Inversion.  Let  P,  Fig.  41,  denote  any  point 
distant  OP  =  r  from  a  fixed  point  0,  and  let  a  point  P'  be  taken 
on  the  line  OP  such  that  OP(=  r)  x  OP'(=  r')  =  R\  Then  P 
and  P'  are  said  to  be  inverse  points  with  respect  to  the  sphere, 
called  the  sphere  of  inversion,  with  center  0,  called  the  center  of 
inversion,  and  radius  R.  P  and  P'  are  also  called  the  geometri- 
cal images  of  one  another  in  the  sphere.  The  process  of  obtain- 
ing P  from  Pf,  or  P'  from  P,  by  the  relation 


Fig.  41. 

OP  OP'  =  R2  or  rrf  =  R2  ( 106) 

is  called  inverting  P  or  Pf  with  respect  to  the  given  sphere. 

If  every  point  of  a  given  surface,  volume,  or  curve  is  inverted 
with  respect  to  a  given  sphere,  a  new  surface,  volume,  or  curve 
will  be  obtained  which  is  called  the  inverse  or  geometrical  image 
of  the  given  surface,  volume,  or  curve  with  respect  to  the  given 
sphere. 

31.  Inverse  of  a  Sphere.  Inverse  of  a  Plane.  The  image  of  a 
sphere  (or  circle)  is  another  sphere  (or  circle),  the  centers  of  the 
two  spheres  (or  circles)  and  of  the  sphere  of  inversion  being  on 
the  same  straight  line.  To  prove  this,  let  C,  Fig.  42,  be  the  center 
of  the  given  sphere  (or  circle)  of  radius  a,  distant  OC—b  from 


112          ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

0,  the  center  of  inversion.  With  O  as  origin  and  OB  as  initial 
line,  the  equation  of  the  given  circle  (or  sphere)  in  polar  coordi- 
nates (r,  0)  is 

r2  —  2br  cos  6  +  b2  —  a1  =  o  (IO7) 

The  equation  of  the  inverse  surface  is  obtained  from  (107) 
by  substituting  for  r  its  equal  R2/rf,  the  coordinates  of  the  in- 
verse being  rf,  0.  Thus  we  find 

r'2  -  2R2bj(b2  -  a2)  •  r'  cos  0  +  R*j(b2  -  a2)  =  o      ( 1 08) 


Fig.  42. 

which  a  comparison  with  (107)  shows  to  be  the  equation  of  a 
sphere  (or  circle)  with  center  at  Cf  distant 

V  =  R2l>/(P-a2)  (109) 

from  0,  and  with  radius 

a'  =  [b'2  -  R*j(b2  -  a2)]  *  =  R2aj(P  -a2)  (no) 

When  the  given  sphere  passes  through  the  center  of  inversion, 
i.  e.,  when  b  =  a,  Fig.  43,  (107)  and  (108)  become 

r—  2acos6  —  o  (IJI) 

and  r'  cos  6  —  R2/2a  =  o  ( 1 1 2) 

(112)  is  the  equation  of  a  plane  (or  straight  line)  distant 

p  =  I?l2a  (113) 

from  0. 


ELECTRIC   FIELDS   AND   CONDENSERS.  113 

Conversely,  (112)  inverts  into  (in),  so  that  the  image  of  a 
plane  (or  straight  line)  is  a  sphere  (or  circle)  passing  through  the 
center  of  inversion  and  with  radius 


Fig.  43. 


(114) 

The  above  propositions  can  also  be  easily  demonstrated  by 
purely  geometrical  methods. 

32.  The  Angle  at  which  Two  Curves  or  Surfaces  Intersect  is  Un- 
altered by  Inversion.  To  establish  this  proposition  for  two  curves 
(which  will  also  establish  it  for  two  surfaces),  let  AB  and  AC, 


Fig.  44. 


Fig.  44,  be  the  elements  of  two  curves  intersecting  in  the  point 
A  at  an  angle  6,  and  A1 Bf  and  Ar  C'  their  inverses  cutting  at  the 
angle  0' '.  The  triangles  AOB  and  A' OB'  are  evidently  similar, 
since  rrf  =  R2,  and  likewise  the  triangles  OA  C  and  OA' C' . 
Hence 

6'  (-  angle  OA' C'  -  angle  OA'B')  =  angle  OAB 
—  angle  OAC  =  6 


114          ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

33.  Electrical  Inversion.  The  electric  potential  at  a  point  B 
due  to  a  charge  dq  at  a  point  A  is  dVb  =  dql^rrcAB,  and  the 
potential  at  B',  the  inverse  of  B,  due  to  a  charge  dq'  at  A',  the 
inverse  of  A,  is  dVb,  =  dq'  j  ^trcA'  B'  .  Thus 

dVb,\dVb  =  dq'jdq  •  ASIA'S' 
But,  if  (9  is  the   center  of  inversion,  ABJA'B'  =  OAJOB'. 

dVb,  =  dVbdq'jdq  •  OAJOB'  (i  1  6) 

If  d/  is  so  chosen  that 

dq'  jdq  =  -RjOA  =  -  Rjr  (117) 

dq'  is  called  the  electric  image  by  inversion  of  dq  with  respect  to 
the  sphere  of  radius  R  and  with   center  0  ;  and  (i  16)  becomes 

dVb,  =  -  dVbRjOBf  =  -  dVb4ircRl4ircOB'         (  1  18) 

If.  now  we  have  any  electric  distribution  whatever,  and  it  pro- 
duces at  B  a  potential  Vb  =  fdVb  ;  and  if  we  place  at  the  geo- 
metrical images  of  the  charged  points  charges  related  to  the 
charges  at  the  original  points  according  to  (117),  i..c.t  form  a 
distribution  which  is  the  electric  image  by  inversion  of  the  orig- 
inal distribution,  the  potential  at  B'  will  be  Vb,  =  §dVb,.  Hence 
(118)  gives  for  the  potential  at  Sf  due  to  the  electric  image  by 
inversion  of  the  original  distribution 


(119) 

which  is  the  potential  which  would  be  produced  at  Bf  by  a 
charge  —  V^qircR  at  Ot  the  center  of  inversion.  If,  therefore, 
we  place  at  (9  a  concentrated  charge  q  =  -f  V^cR,  the  point 
B1  will  be  at  zero  potential  in  the  presence  of  this  charge  to- 
gether with  the  inverted  system.  If  the  potential  Vb  =  V  is  the 
same  for  all  charged  points  of  the  original  system,  that  is,  if  the 
original  charge  is  distributed  over  an  equipotential  surface,  as  the 
surface  of  a  conductor,  at  potential  V,  the  potential  at  a  point  Sf 
of  the  inverse  surface  due  to  the  inverse  distribution  will  be 


ELECTRIC    FIELDS   AND    CONDENSERS.  11$ 


Hence  if  a  charge  q  =  + 
placed  at  0,  the  potential  at  all  points,  such  as  B'  ',  of  the  inverse 
surface  will  be  zero.  The  introduction  of  q  does  not  alter  the 
distribution  upon  the  inverse  surface,  but  renders  this  surface 
equipotential  so  that  it  may  be  made  conducting  without  disturb- 
ing the  distribution.  The  electric  field  after  the  introduction  of 
q  is  the  field  bounded  by  a  concentrated  charge 

q  —  47rcR  V  (  1  20) 

at  0  and  the  inverse  of  the  original  surface  at  zero  potential. 

Conversely,  if  we  have  a  charge  q  concentrated  at  a  point  0 
in  the  presence  of  a  charged  surface  at  zero  potential,  we  can  in- 
vert the  surface  and  its  distribution  (not  including  the  charge  q) 
with  reference  to  a  sphere  of  radius  R  and  center  0  and  obtain 
the  inverse  surface  charged  to  a  uniform  potential 

V=ql4ircR  (121) 

alone  in  the  field,  the  charge  q  being  annulled. 

Or,  if  we  have  a  surface  at  zero  potential  in  the  presence  of  a 
charge  qt  concentrated  at  a  point  0,  and  its  image  system  (on  the 
other  side  the  surface),  we  can  invert  the  image  system  (not  in- 
cluding the  charge  q)  and  the  given  surface  with  respect  to  a 
sphere  of  radius  R  with  center  0,  and  as  a  result  obtain  the  in- 
verse surface  at  the  uniform  potential 


and  within  (or  on  one  side)  the  inverse  image  system  which  pro- 
duces on  the  other  side  the  same  field  as  that  connected  with  the 
surface  itself  when  charged  to  uniform  potential  V.  From  the 
image  system  the  distribution  upon  the  surface,  when  charged  in 
this  manner,  or  made  conducting,  can  readily  be  found. 

The  direct  inversion  of  the  distribution  on  the  surface  at  zero  po- 
tential would  be,  in  general,  a  difficult  matter.  Hence  the  second 
of  the  two  processes  of  inversion  just  described  is  usually  preferable. 

34.  The  Electric  Surface  Density  upon  the  Inverse  Surface. 
The  electric  surface  density,  a',  at  any  element  dSf  of  the  inverse 


Il6          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

i 

surface  corresponding  to  the  element  dS,  with  density  <r,  of  the 
original  surface,  can  easily  be  found  in  terms  of  <r,  R,  and  rf,  the 
distance  from  0  to  dS' '.  For  we  have 

c'dS'lvdS  =  dq'ldq  =  -  Rjr,  or  <r'/a  =  -R/r-  dSjdS' 
But  dS  is  similar  to  dSt'  hence  dSjdS'  =  r2/^2  and 

<r'/<r  =  -  Rr/r'2  =  -  R*jrfZ 
Thus  af  =  -  aR3/r'*  (122) 

As  stated  in  §  33,  <rf  is  not  altered  by  the  introduction  of  the 
charge  q  at  0. 

35.  The  Sphere  and  Plane.  Consider  a  sphere  of  radius  a  uni- 
formly electrified  to  potential  V.  Let  the  sphere  be  inverted 
with  respect  to  a  sphere  of  radius  R  with  its  center  0  upon  the 
surface  of  the  given  sphere.  The  sphere  inverts  into  an  infinite 
plane  (§31)  distant/  =  R2/2a  from  O.  If  now  we  place  at  0  a 
charge  q  =  qircR  V,  the  plane  will  be  at  zero  potential  in  the 
presence  of  the  charge  q.  The  electric  surface  density  over  the 
given  sphere  was  uniform  and  equal  to  <r  =  qirca  Vj^Tra2  =  cVja. 
Hence  the  density  at  a  point  on  the  plane  distant  r'  from  0  is 

<r'  =  —  aR3/r'3  =  -  gR2/47rar'3  =  -  tf/2'irr'*        (123) 

which  is  the  result  obtained  in  §  15,  (40),  proper  attention  being 
paid  to  sign. 

Conversely,  we  may  start  with  the  plane  at  zero  potential 
electrified  to  density  <rf  =  —  pq'^irr'*,  and  by  inversion  obtain 
the  distribution  upon  a  freely  electrified  sphere  of  radius  a.  Thus 
the  plane  inverts  into  the  sphere,  and  the  image  of  q  in  the  plane, 
viz.,  —  q  distant  /  from  the  plane  on  the  other  side  from  q,  in- 
verts into  the  charge  -f  RqJ2p  at  the  center  of  the  sphere  ;  and 
this  brings  the  sphere  to  the  potential  RqJ2p7r^ca  =  qjqircR  =  V. 

Next  let  the  center  of  inversion  be  taken  outside  the  sphere  or 
inside  the  sphere,  and  let  the  sphere  of  inversion  be  so  chosen 
that  the  given  sphere  inverts  into  itself.  This  makes  R2  =  x2  —  a2 
when  the  point  0  is  outside  the  sphere,  and  R2  =  a2  —  x2  when 


ELECTRIC    FIELDS   AND    CONDENSERS. 


117 


0  is  inside,  if  x  denotes  the  distance  from  0  to  the  center  of  the 
sphere.  Thus  the  surface  density  at  a  point  of  the  sphere  dis- 
tant r1  from  0,  when  a  charge  q  =  ^ircRV^  4.7rc(x* 
is  placed  at  0  is 


=  -  q(xi 


(I24) 


in  accord  with  (83)  and  (84). 

Conversely,  we  may  pass  at  once  from  this  distribution  to  that 
upon  the  isolated  sphere. 

36.  Two  Spheres  Intersecting  at  Right  Angles.  The  field  of 
§  29  may  also  be  obtained  by  inverting  the  distribution  upon  two 
infinite  planes  meeting  at  right  angles  and  at  zero  potential  under 


Fig.  45. 


the  influence  of  a  charge  q  =  ^rrcRV 'at  a  point  P  distant  a  and  b 
therefrom  (Fig.  45). 

Let  the  planes  and  the  image  system  in  the  planes  of  the 
charge  q  at  Pbe  inverted  with  respect- to  a  sphere  with  center  P 
and  radius  R.  The  two  planes  invert  into  two  spheres  intersect- 
ing at  right  angles  (corresponding  parts  of  surfaces  are  shown 


n8 


ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 


in  full  or  dotted  lines),  and  of  radii  a  =  R2/2a  and  0  =  R2/2&, 
respectively.  The  images  of  q  at  Pin  vert  into  charges  qRJ2a 
=  gajR  at  A9  qRJ2b  =  qfijR  at  C,  and  -  qR\2(c?  +  tf}\ 
=  —  ga,p/R(a?  +  y32)*  at  B.  And  in  the  presence  of  these 
charges  the  two  spheres  are  at  the  potential  V=  qj^TrcR,  and 
the  field  outside  the  spheres  is  the  field  required.  The  total 
charge  upon  the  spheres,  when  made  conducting,  the  capacity, 
etc.,  may  now  be  found  as  in  §  29. 

By  inverting  the  system  consisting  of  two  planes  meeting  at 
the  angle  TT//Z,  etc.,  §  28,  the  field  surrounding  two  spheres  cut- 
ting at  that  angle  may  be  obtained.  When  n  —  infinity,  this 
problem  merges  into  that  of  the  next  article. 

37.  Two  Spheres  in  Contact.  By  inverting  the  two  planes  of 
§  28  with  respect  to  a  sphere  of  radius  R  and  center  P  we  ob- 


Flg.  46. 

tain  two  spheres  of  radii  A  =  R2/2a  and  B  —  R2J2b  in  contact  at 
P,  Fig.  46.  All  the  images  to  the  right  of  P  invert  into  the 
region  within  the  sphere  B  and  all  to  the  left  of  P  into  the  region 
within  the  sphere  A  ;  and  the  system  of  two  spheres  in  contact 
is  brought  to  the  uniform  potential  V  =.  qj^TrcR,  and  may  be 


ELECTRIC    FIELDS   AND    CONDENSERS.  119 

made  conducting,  and  the  inner  field  destroyed,  without  affecting 
the  external  field. 

In  the  original   system  the  distances  from  P  of  the  positive 

charges,  +  q,  are 

2d,  Afd,  6d,  •  •  •  to  the  right 
and 

2d,  4d,  6d,»>  to  the  left 

and  the  distances  from  P  of  the  negative  charges,  —  q,  are 

2b,  2b  +  2d,  2b  +  4^,  •  •  •  to  the  right 
and 

2a,  2a  +  2d,  2a  -\-4-d,  •  •  •  to  the  left 

This  system  inverts  into  the  system  of  negative  charges 

—  RqJ2d,  —  Rqj^d,  —  Rqj6d,  -  -  •  within  the  sphere  B 
and 

—  RqJ2d,  —  RqJ4d,  —  Rqj6d}  .  .  .  within  the  sphere  A 

and  the  system  of  positive  charges 

RqJ2b,  Rqj(2b  +  2d\  Rqj(2b  +  4^),  .  .  .  within  the  sphere  B 
and 
RqJ2a,  Rqj(2a  +  2d},  Rqj(2a  -f  4^),  •  •  .  within  the  sphere  A 

The  total  charge  of  the  images  within  the  sphere  A,  or  the 
total  charge  upon  the  surface  of  A  when  made  conducting,  is 

-{\_l/a+  i/(a  +  <i)+  I  /(a  +  2d)  +  •  .  •] 


and  that  of  the  images  within  B,  or  upon  the  surface  of  the 
sphere  B  when  a  conductor,  is 

qb  =  Rqj  2  •  {  [  I  Ib  +  I  l(b  +  d  )  +  I  j(b  +  2d)  +  -  -  -] 

-(ljd+  lJ2d+ 
Now 


120          ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 


_  +  i)] 

Hence  »=o 


/  \B  +  n(A  +  ^)  (»  +  i)] 
S   l/[^+K^  +  ^)(«+  0] 

n--0 

Interchanging  A  and  ^,  we  obtain 
qb  =  47rcV&Al(B  +  A)  ^i/[A  +  *(£  +  ^)  (»  H-  i)]      (126) 

n=0 

The  capacity  of  the  two  spherical  conductors  in  contact  is 

S=qlV=(qa+qMV  (127) 

We  shall   consider  further   two  particular  cases  :    (i)    when 
A  =  B,  (2)  when  BjA  is  very  small. 
(i)  A  =  B.     In  this  case 


i  /(i  +  2»)  (»  H-  i) 

1/3-4+  i/S-6  +  ..-)  (I28) 

=  47TcA  Flog  2  =  477-^  F  x  0.693 

and  5  =  2qJ  F=  STrcA  log  2  =  qircA  x  1.386  (129) 

Thus  the  capacity  of  the  system  of  two  equal  spheres  in  contact 
is  equal  to  1.386  times  the  capacity  of  a  single  isolated  sphere 
of  the  same  radius. 

The  energy  of  the  field  surrounding  the  spheres  is 

W=  \q  F=  47rcA  log  2  •  F2  =  g2/i67rcA  log  2        (130) 


ELECTRIC    FIELDS    AND    CONDENSERS.  121 

From  this  expression  and  (  1  1  )  it  is  easy  to  compute  the  work 
done  against  the  electrical  forces,  or  the  work  done  upon  the 
electric  field,  when  two  equal  spheres  with  equal  charges  are 
brought  together  from  an  infinite  distance  (or  a  great  distance, 
practically)  apart. 

(2)  BjA  very  small.  When  BjA  =  o,  the  sphere  A  is  alone 
in  the  field  at  potential  V\  hence  qa  —  ^nrcA  V,  a  relation  which 
holds  approximately  when  BjA  is  very  small.  In  this  case  we 
have  for  qb,  approximately, 


i/22+  •••)  =  47rc£2/A-7T*/6-  V  (131) 

The  capacity  of  the  system  is,  approximately, 

S  =  47rc(A  +  7r2/6<B2/A)  (132) 

The  electric  surface  density  upon  the  larger  sphere,  except 
near  the  point  of  contact,  is,  approximately, 

'.-'VIA  (i33) 

while  the  average  density  upon  the  smaller  sphere  is,  approxi- 
mately, 

(I34) 


The  relation  <rb/<ra  =  7r2/6  will  hold  very  approximately  when 
the  small  sphere  B  touches  any  conducting  surface  A  which,  like 
a  large  sphere,  is  nearly  plane  in  the  neighborhood  of  the  point 
of  contact,  the  nearly  uniform  density  in  that  vicinity  being, 
before  contact  with  the  small  sphere,  aa. 


CHAPTER    III. 

STANDARD    CONDENSERS.     CONDENSER   SYSTEMS. 
ELECTROMETERS. 

1.  Actual  Condensers.  The  electric  fields  and  condensers  or 
leydens  discussed  in  Chapter  II.  are  ideal,  the  conditions  assumed 
being  impossible  to  realise  completely  in  practise.  Concentrated 
charges  and  infinite  conductors  do  not  exist;  one  or  two  con- 
ductors cannot  be  infinitely  removed  from  all  other  conductors ; 
and  all  the  tubes  from  the  first  conductor  will  not,  in  general, 
terminate  upon  the  second,  unless  the  second  conductor  com- 
pletely surrounds  the  first  or,  what  amounts  to  the  same  thing,  is 
connected  with  the  walls  of  the  room,  which  thus  becomes  one, 
electrically,  with  the  second  conductor.  The  field  in  this  latter 
case  cannot  be  rigorously  computed,  however;  and  though  it  is 
possible  to  construct  a  condenser  of  concentric  spheres  with  a 
high  degree  of  accuracy,  or  of  conductors  of  other  form  so 
arranged  that  one  completely  surrounds  the  other,  and  such  that 
the  electric  field  can  be  rigorously  computed,  yet,  to  use  the 
system,  an  insulated  wire  must  pass  through  an  opening  in  the 
outer  conductor  to  the  inner,  and  through  this  opening,  however 
small,  some  tubes  will  emerge  and  pass  to  the  external  surface 
of  the  outer  conductor  and  to  other  bodies.  The  wire  and  the 
opening  also  disturb  the  field  in  other  ways. 

But  finite  portions  of  all  the  fields  can  be  very  nearly  pro- 
duced, and  the  results  established  above  for  ideal  condensers  and 
fields  can  be  applied  without  sensible  error  to  actual  systems. 
This  can  be  done,  for  example,  with  the  systems  consisting  of 
two  parallel  similar  conducting  surfaces,  as  two  spheres,  two 
planes,  or  two  cylinders,  by  making  the  distance  between  them 

122 


CONDENSERS   AND    ELECTROMETERS. 


123 


small  in  comparison  with  their  linear  dimensions.  Then  the  part 
of  the  field  between  the  surfaces  —  concentric  spheres,  coaxial 
cylinders,  or  parallel  planes  —  becomes  practically  identical,  ex- 
cept near  the  edges,  with  those  already  described,  and  the  part 
of  the  field  outside  this  region  relatively  very  weak.  The  capacity 
of  such  a  condenser,  of  any  form,  is  approximately  the  product 
of  the  area  of  one  of  its  conductors  by  the  permittivity  of  its 
dielectric  divided  by  the  distance  between  the  conductors,  the 
intensity  being  practically  constant  in  magnitude  throughout  the 
dielectric. 

A  plane  section  of  the  tubes  of  displacement  of  a  square  par- 
allel plate  condenser  taken  parallel  to  one  edge  and  perpendicu- 
lar to  the  plates  through  their  centers  is  shown  in  Fig.  47.  The 


Fig.  47. 

tubes  in  the  neighborhood  of  the  section  are  supposed  to  be 
formed  by  moving  the  diagram  perpendicularly  to  its  plane.  The 
diagram  is  drawn  only  approximately.  The  tubes  are  closely 
concentrated  and  uniformly  distributed  between  the  plates,  except 
near  the  edges,  where  the  field  becomes  less  intense,  and  sparsely 
distributed  over  the  outer  surface,  becoming  less  numerous  as 
the  central  points  A  and  B  are  approached.  These  results  fol- 
low from  the  principle  of  symmetry  and  the  fact  that  the  voltage 
J  EdL  is  the  same  along  every  line  of  intensity  from  one  con- 
ductor to  the  other,  which  makes  the  average  intensity  great 
where  the  length  of  the  line  is  small,  and  vice  versa. 


124          ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

As  the  capacities  of  ordinary  condensers  are  computed  only 
roughly  for  construction  purposes,  and  then  determined  or  ad- 
justed accurately,  when  necessary,  by  comparison  with  standard 
capacities,  it  is  of  little  importance  whether  their  fields  are  such 
that  the  capacities  can  be  determined  accurately  by  geometrical 
measurement  or  not.  But  this  is  obviously  necessary  in  the  case 
of  condensers  designed  as  absolute  standards  of  capacity.  Such 
standards  have  been  constructed  of  concentric  spheres,  coaxial 
cylinders,  and  parallel  plates.  The  first  form  does  not  need 
further  description  here ;  the  last  two  will  be  described  in  the 
next  article. 

To  eliminate  the  electric  field  surrounding  the  earth,  all  the 
apparatus  here  described  will  be  supposed  enclosed  within  a 
hollow  conductor,  such  as  the  walls  of  a  room  in  a  house.  The 
phenomena  would  not  be  essentially  different,  however,  outside 
such  an  enclosure.  The  potential  of  the  walls  of  the  enclosure 
(earthed)  will  be  assumed  zero. 

2.  The  Standard  Parallel  Plate  Condenser.  The  construction 
of  this  condenser  and  its  electric  field  near  the  center,  drawn 
like  the  field  in  §  I,  are  shown  in  Figs.  48  and  49.  A  and  A' 


HP  A C  B          C A | 

r<* 

•A'  A' 

Fig.  48. 

are  the  two  plates  distant  d  apart.  The  central  portion,  B,  of  A 
is  separated  from  the  rest  by  an  air  gap  CC  whose  breadth  is 
very  small  in  comparison  with  d.  Above  and  continuous  with 
the  plate  A  is  a  metal  cover  D,  which  forms  with  A  and  B  a 
hollow  conductor  closed  except  for  the  gap  CC.  If  B  is  put  in 
metallic  contact  with  A,  and  the  condenser  then  charged,  the 
electric  field  shown  in  the  figures  will  result,  and  will  remain 
when  B  is  again  insulated  from  A.  Since  the  region  above  B  is 
the  interior  of  a  hollow  conductor  practically  closed,  all  the 
tubes  from  B  will  proceed  to  the  upper  surface  only  of  A' ,  and 


CONDENSERS    AND    ELECTROMETERS. 


125 


but  few  tubes  will  emanate  from  the  upper  surface  of  B  and  pass 
through  the  gap  CC  to  A' .  The  field  below  B,  being  remote 
from  the  edges,  will  be  sensibly  uniform  except  in  the  immediate 
neighborhood  of  the  gap  C,  which,  however,  will  not  sensibly 
disturb  the  uniform  field  near  A' .  By  symmetry,  sensibly  half 


Fig.  49. 
(The  width  of  the  gap  Cis  greatly  exaggerated.) 

the  tubes  which  terminate  upon  the  small  area  of  Ar  just  beneath 
the  gap  CC  must  emanate  from  B  and  half  from  A.  If  b  de- 
notes the  area  of  B,  and  a  that  of  CC,  the  area  of  A'  receiving 
tubes  from  B  is  thus 

A  =  b  +  \a  (i) 

The  charge  upon  B  is  therefore  the  same  as  the  charge  upon 
the  area  A  =  b  +  \a  of  Af,  which  is  the  same  charge  B  would 
have  if  its  area  were  A  =  b  +  \a  and  there  were  no  gap.  The 
capacity  of  the  condenser  formed  by  B,  A',  and  the  tubes  con- 
necting them  is  therefore 


The  conductor  A  which  surrounds  B,  and  by  means  of  which 
the  field  beneath  B  is  made  uniform,  is  called  &  guard  ring. 


Fig.  50. 


The  force  of  attraction  between  B  and  A'  is  the  force  acting 
upon  the  area  A  =  b  +  \a  of  A'  \  that  is 


126          ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

F=  \EDA  =  \c&A  =  \c  V*jd*  •  A  (3) 


if  V  denotes  the  voltage  between  the  plates  AB  and  A  '. 

The  Standard  Cylindrical  Condenser.  The  guard  ring  con- 
struction can  be  applied  equally  well  to  the  condenser  formed  of 
coaxial  cylinders.  The  construction  of  the  condenser  is  shown 
in  Fig.  50,  the  inner  cylinder  serving  also  as  guard  ring  and  pro- 
tector. 

3.  Electroscopes  and  Electrometers.     An  electrometer  is  an  in- 
strument for  measuring  voltages,  or  electric  potential  differences, 
by  means  of  the  forcive  acting  between  electrified  bodies.      Other 
instruments  for  measuring  voltages  will  be  described  later.     An 
electroscope  is  a  crude  electrometer,  used  principally  for  detect- 
ing rather  than  measuring  electrical  effects. 

4.  The  Kelvin  Absolute  Electrometer.     This   instrument  con- 
sists of  a  condenser  constructed  like  the  standard  parallel  plate 
condenser  of  §  2  with  certain  modifications  and  additions  :  The 
plate  B  (in  one   of  the  commonest  forms  of  the  instrument)  is 
connected  through  a  small  opening  in  the  sheath  D  with  one 
arm  of  a  gravity  balance,  so  that  the  force  F  between  B  and  A9 
can  be  determined  by  weighing.     A  vertical  micrometer  screw 
topped  by  an  insulating  support  which  carries  Af  enables  the 
distance  d  to  be  varied  and  measured.     B  is  kept  in  electrical 
connection  with  A,  and  an  optical  device  is  provided  with  the 
aid  of  which  the  planes  of  A  and  B  are  always  made  coincident 
(by  altering  the  weights  in  the  balance  pans  or  by  varying   the 
distance  d)  when  the  force  upon  B  is  to  be  measured. 

To  measure  a  voltage,  A  and  A'  are  first  metallically  con- 
nected, and  the  weights  on  the  balance  pans  adjusted  until  the 
planes  of  A  and  B  are  coincident.  Then  the  connection  between 
A  and  A1  is  broken,  and  they  are  brought  to  the  difference  of 
potential  V,  to  be  determined.  The  force.  -F,  due  to  the  attrac- 
tion between  the  plates  is  balanced  by  the  addition  of  known 
weights,  and  the  distance  d\s  measured.  Then  by  (3) 


CONDENSERS    AND    ELECTROMETERS. 


whence  F=  d  \/2F/cA  (4) 

The  above  method  of  using  the  electrometer  is  called  the 
idiostatic  method,  as  the  voltage  to  be  determined  is  the  only  one 
employed.  Since  F  is  proportional  to  the  square  of  the  voltage, 
alternating  as  well  as  direct  voltages  can  be  measured. 

In  the  heterostatic  method  an  auxiliary  agent  with  a  constant 
voltage  V  is  also  used.  When  this  voltage  alone  is  applied  to 
the  electrometer  terminals,  we  have  from  the  last  equation 

Vr  =  d'V2F[cA 

if  F  denotes  the  force  upon  B  when  the  distance  between  the 
plates  is  d'  . 

If  now  we  connect  up  in  series  the  agent  whose  voltage  V  is 
to  be  determined  and  the  agent  whose  voltage  is  V1  ',  both 
voltages  being  directed  in  the  same  way  so  that  the  resultant 
voltage  is  V  -\-  F',  we  have,  when  the  voltage  F+  V  is  applied 
to  the  electrometer  terminals, 


Vf  =  d 

if  dn  denotes  the  distance  between  the  plates  when  the  force  F 
remains  the  same  as  before. 

Subtracting  the  first  equation  from  the  second  gives 

V=(d"-d')V2FI~tA  (5) 

The  advantage  which  this  method  has  over  the  other  is  due  to 
the  much  greater  accuracy  with  which  the  micrometer  permits 
the  measurement  of  the  difference  of  the  two  distances  d"  —  d1 
than  either  separately. 

Since  F  is  proportional  to  F2,  it  becomes  so  small  for  small 
voltages  that  it  cannot  be  accurately  measured  with  this  instru- 
ment. This  form  of  absolute  electrometer  is  therefore  used  only 
for  measuring  large  potential  differences,  and  small  voltages  are 


128 


ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 


measured  by  the  quadrant  electrometer,  the  Bichat  and  Blondlot 
electrometer,  the  capillary  electrometer  of  Lippmann,  or  a  form 

of  absolute  electrometer  recently  de- 
vised by  Perot  and  Fabry.  The  first 
two  instruments  are  described  in  the 
succeeding  articles. 
"  Bichat  and  Blondlot 's  Electrometer 
(Modified).  This  instrument  (Fig.  51) 
consists  of  a  metallic  circular  cylin- 
der C  suspended  from  the  arm  of  a 
balance  (or  connected  to  another 
•dynamometer)  by  a  fine  wire  DE, 
with  its  axis  vertical  and  coincident 
with  the  axis  of  a  longer  hollow  me- 
tallic circular  cylinder  AH,  cut  in  two 
at  FG  and  projecting  well  beyond  the 
ends  of  C,  the  difference  between  the 
radii  (Zt  and  L2,  L^  >  L2)  of  the  two 
cylinders  being  small  in  comparison 
with  either  radius. 

Let  A,  B  and  C  be  charged,  the 
voltage  from  B  to  A  being  denoted 
by  VBA,  that  from  C  to  A  by  VA, 
and  that  from  C  to  B  by  VB,  C  being 
charged  by  the  wire  DE,  dipping  in  a 
conducting  liquid.  The  field,  in  plane  section  through  the  axis,  for 
the  case  in  which  VA  >  VB  and  VBA=  VA—VB  therefore  positive, 
is  shown  approximately  in  the  figure  (all  the  lines  of  intensity 
should  touch  the  conductors  normally).  Except  near  and  beyond 
E  and  Ht  and  G  and  Ft  the  field  is  cylindrically  radial,  and  its 
capacity  per  unit  length  is  constant  and  equal  to 

5  =  2TTC  -  log  LJL2 

By  symmetry,  there  is  no  resultant  horizontal  force  acting  on 
C.     In  general  the  vertical  forces  acting  on  C  at  H  and  E  are 


Fig.  51, 


CONDENSERS   AND    ELECTROMETERS.  129 

not  equal  and  opposite.  The  resultant  force  can  be  found  as 
follows  :  Imagine  C  to  be  moved  downward  an  infinitesimal  dis- 
tance dx.  The  capacity  of  the  condenser  AC  is  increased  by 
Sdx  and  that  of  the  condenser  BC  by  —  Sdx,  no  sensible  change 
occurring  in  the  capacity  of  the  non-cylindrical  parts  of  the  field. 
The  increase  in  the  energy  of  the  field,  if  the  voltages  are  kept 

constant,  is 

-  VB2) 


Hence,  by  §  55,  1.,  the  resultant  force  acting  downward  upon  C  is 

F-  dwdx=  s  v;  -  v/)  =  ±SVBA(  VE  +  VA} 


(1)  If  Fg  is  great  in  comparison  with  VABt  the  voltage  to  be 
measured,  this  equation  becomes,  with  a  negligible  error, 

F=SVBA-VB  (7) 

Hence,  if  VB  is  kept  constant  and  VBA  varied,  F  is  proportional 
to  VBA. 

(2)  If  A  and  B  are  connected  to  the  terminal  plates  of  an 
auxiliary  battery  consisting  of  an  even  number  of  similar  voltaic 
cells  in  series,  and  if  one  terminal  of  the  cell,  condenser,  or  other 
agent  whose  voltage  V  is  to  be  measured  is  connected  to  the 
central  point  of  this  auxiliary  battery,  the  other  to  the  conductor 
C,   we  have,  if  ^  denotes  the  e.m.f.   of  the  auxiliary  battery, 
VBA  =  ¥,  VB  =  V-  ¥*  >  VA  =  F+  J¥  ;  and  (6)  becomes 

F=  SV  -  V  (8) 

so  that  F\s  proportional  to  V\{  "SP"  is  kept  constant. 

(3)  If  B  and  C  are  connected  together,   VB  =  o,    VA=  VBAt 
and  (6)  becomes 

(9) 


Equations  (7),  (8),  and  (9)  indicate  three  methods  of  com- 
paring voltages  with  the  instrument,  the   force  F  being   meas- 


130 


ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 


ured  with  the  balance  (or  other  form  of  dynamometer).  If  5  is 
determined  from  direct  measurement,  and  F  measured  in  dynes, 
the  third  method  gives  an  absolute  determination  of  the  voltage 
VBA.  The  first  and  second  methods  are  called  heterostatic y  as  an 
auxiliary  voltage,  VB  or  ¥",  is  employed  in  addition  to  that  to  be 
measured.  The  third  method  is  called  idiostatic,  since  the  volt- 
age to  be  determined  is  the  only  one  applied. 

Jtf.  The  Kelvin  Quadrant  Electrometer.     This  instrument  (Fig. 
52)  is  constructed  as  follows  :  A  right  circular  cylindrical  me- 

OR 


< 

)  M 

W 

I    1 

H2S04 

| 

Fig.  52. 

tallic  box,  with  its  axis  vertical,  is  cut  symmetrically  into  four 
quadrants  At  A',  B,  B1 ,  separately  insulated  on  glass  rods,  but 
connected  by  wires  in  pairs,  A  to  A!  and  B  to  B' ,  so  that  when 
the  field  is  static  there  is  never  a  potential  difference  between  op- 
posite quadrants.  A  light  aluminium  needle  C,  consisting  of 
two  equal  opposite  flat  quadrantal  arcs  CC  and  C'  C'  attached  by 
thin  radii  at  their  extremities  to  a  central  vertical  rod  R,  is  sus- 
pended from  a  support  by  two  silk  fibers  (or  other  insulating 
torsion  device)  in  such  a  way  that  the  arcs  CC  and  C'  C'  are 
horizontal,  concentric  with  the  quadrant  cylinder,  and  midway 
between  the  top  and  bottom  of  the  box.  When  the  quadrants 
and  the  needle  are  all  connected  together,  so  that  there  is  no 
potential  difference  between  any  two  parts  of  the  system,  the  arcs 


CONDENSERS    AND    ELECTROMETERS.  131 

C,  C  are  adjusted  to  lie  symmetrically  with  respect  to  the  two 
quadrant  pairs  AA'  and  BB' ',  as  shown  in  the  figure.  To  the 
rod  R  is  attached  a  mirror  M,  by  means  of  which  and  a  lamp 
and  scale  or  telescope  and  scale  any  deflection,  6,  of  the  needle 
can  be  read,  and  on  the  other  side  of  the  quadrants  a  vertical 
platinum  wire  Wy  ending  in  a  platinum  vane  V.  The  end  of  the 
wire  and  the  vane  hang  free  in  dry  sulphuric  acid  contained  in  a 
glass  vessel  G,  the  outer  surface  of  which  is  partly  covered  with 
tin  foil.  The  sulphuric  acid  serves  to  make  electrical  contact 
with  the  needle,  to  dampen  the  needle's  motion,  and  to  form  with 
the  tin  foil  and  glass  vessel  a  condenser  of  considerable  capacity, 
whose  function  is  to  keep  constant  the  potential  difference  be- 
tween the  needle  and  the  case.  The  whole  instrument  is  enclosed 
in  a  tight  case,  often  an  extension  of  the  vessel  G  (whose  tin  foil 
covering  is  then  outside)  and  is  kept  dry  by  the  sulphuric  acid 
within.  The  case,  largely  metal,  serves  also  to  screen  the  needle 
and  quadrants  from  any  external  field. 

If  the  instrument  is  symmetrically  made  and  adjusted,  the 
arcs  CC  and  C'  C'  form  with  the  two  quadrant  pairs  A  A'  and 
BB'  two  condensers,  the  capacity  of  each  of  which,  per  unit 
angle  subtended  at  the  center  of  the  system,  is  the  same,  let 
us  say  S,  and  constant,  except  near  the  edges  of  the  arcs 
and  quadrants,  for  all  but  exceedingly  large  deflections  of  the 
needle. 

Also,  if  the  instrument  is  symmetrically  made  and  in  adjust- 
ment, the  needle  will  obviously  not  be  deflected,  even  when 
charged,  as  long  as  the  quadrants  are  all  connected  together. 
If  the  needle  and  the  quadrant  pairs  A  A '  and  BBr  are  charged, 
the  needle  will,  in  general,  be  deflected,  coming  to  rest  when  the 
angle  of  deflection,  6,  is  such  that  the  torque  T  upon  it  due  to 
the  electrical  stresses  is  balanced  by  the  return  torque  due  to  the 
twist  of  the  suspension.  To  find  the  relation  between  the  deflec- 
tion and  the  voltage,  we  may  proceed  as  follows,  using  the 
method  of  §  55,  I. 


132          ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

Let  VAB  denote  the  voltage  from  the  quadrants  AA'  to  the 
quadrants  BB'  ',  VA  the  voltage  from  the  needle  to  AA  ,  and  VB 
the  voltage  from  the  needle  to  BB'  . 

When  6  is  increased  by  an  amount  dQ,  the  capacity  of  the  con- 
denser formed  by  CC  f  with  A  A'  is  increased  by  Sdd,  and  that 
of  the  condenser  formed  by  CC'  with  BBf  is  decreased  by  the 
same  amount.  The  increase  in  the  energy  of  the  two  condensers 
is  then 


*  -  \SdQ  V*  =  \SdQ(  VB  -  VA)(  VB  +  VA) 

VA)  =  SdO  •  VAB(VB-\VAB)  =  TdQ 
=  K6dO 

since  VB=^VA-\-  VAB,  and  since  TdO  =  KQ  dQ  is  the  work  done 
in  twisting  the  bifilar  (or  other)  suspension  through  the  angle  dQ 
by  the  torque  T  of  the  electrical  forces,  K  being  the  constant  of 
torsion  of  the  suspension.  The  last  equation  gives 

T/K=e=  *  SIK  •  VAB(  VB  +  VA)  -  SjK  •  VAB(  VB  -  J  VAB}  (10) 

(1)  If  VA  and  VB  =  VA  +  VAB  are  very  large  in  comparison  with 
VAB,  the  voltage  to  be  measured,  VAB  may  be  neglected  without 
appreciable  error  in  the  expression  (  VB  —  ^  VAB)>  an<^  ^  'ls  sensibly 
proportional  to  VAB  and  to  VB.     Hence  by  making  VB  large,  even 
small  potential  differences  VAB  may  be  measured  with  accuracy. 
In  this  case  (10)  becomes 

vAB  =  Kisvs-e  (it) 

(2)  If  the  needle  is  in  metallic  contact  with  one  of  the  quad- 
rant pairs,  as  A  A  ',  VA  =  o,  VB  =  VAB,  and  (10)  becomes 

vAi  =  2Kjs.e  (12) 

Since  the  deflection  in  this  case  is  proportional  to  the  square 
of  the  voltage,  alternating  as  well  as  steady  voltages  can  be 
measured  ;  but  low  voltages,  either  steady  or  alternating,  cannot  be 
measured  with  accuracy  (except  with  very  sensitive  instruments). 

(3)  The  quadrant  pairs  A  A  '  and  BB'  are  connected  to  the  ter- 
minal plates  of  an  auxiliary  voltaic  battery  consisting  of  an  even 
number  of  similar  cells  in  series,  and  one  pole  of  the  voltaic  cell 


CONDENSERS    AND    ELECTROMETERS. 


133 


or  other  agent  whose  voltage  V  is  to  be  measured  is  connected 
to  the  needle  C,  the  other  to  the  central  point  of  the  auxiliary 
battery.  Then,  if  M*  denotes  the  e.m.f.  of  the  auxiliary  battery, 


V 


AB 


V  =  V- 


VB  =  V+  y&  ;  and  (10)  becomes 


(13) 

In  this  arrangement  the  deflection  is  accurately  proportional  to 
V,  whether  Kis  large  or  small  in  comparison  with  "^P. 

The  first  and  third  methods  of  using  the  instrument,  in  which 
a  supplementary  voltage  is  employed  in  addition  to  that  to  be 
measured,  are  called  heterostatic  methods  ;  the  second  is  called 
idiostatic. 

While  the  quadrant  electrometer  cannot  be  used  for  absolute 
measurements,  the  factor  multiplying  0  being  impossible  to  de- 
termine with  accuracy  directly,  this  factor  can  be  determined  in 
any  case  by  measuring  the  deflection  produced  by  a  known  volt- 
age, such  as  that  of  a  standard  cell. 

7.  Condensers  in  Multiple.  When  any  number  n  of  condensers 
whose  separate  capacities  are  Sv  S2,  -  •  •  ,  Sn  are  connected  in 


b. 

Fig.  53. 


multiple,  as  in  Fig.  53,  a,  a  compound  condenser  is  formed  whose 


capacity  is 


(14) 


134          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

provided  that  the  field  of  each  condenser  is  included,  practically, 
between  its  plates  only,  and  therefore  does  not  affect  appreciably 
the  fields  of  the  other  condensers. 

For  if  Fis  the  common  voltage  between  the  separate  pairs  of 
plates  of  the  compound  condenser,  q  the  total  charge  on  each 
compound  plate,  and  qv  qv  -  -  •  ,  qn  the  charges  on  the  separate 
plates  when  the  condensers  are  charged  separately  to  the  volt- 
age Vt  we  have,  for  such  a  system, 

=       +       +  ----  1-  and 


8.  Condensers  in  Series,  When  n  condensers  of  individual 
capacities  Sv  S2,  etc.,  are  connected  up  in  series,  as  in  Fig.  53,  b, 
a  compound  condenser  is  formed  of  capacity 

5-i/(i/5,+  ,/S,  +  •  •  •  +  i/SJ  (15) 

provided  that  the  plates  of  each  condenser  are  so  close  together 
that  sensibly  all  the  tubes  from  one  plate  terminate  upon  the 
other. 

For  if  Fis  the  total  potential  difference  between  the  terminal 
plates  of  the  compound  condenser,  q  the  (numerical)  charge  on 
each  of  them,  and  gv  q2J  etc.,  and  Vv  Vv  etc.,  the  charges  and 
voltages  of  the  individual  condensers,  we  have 

9  =  ft  =    ft  =  ft  =    '  '  '  =  ft 

since  the  intermediate  plates  are  all  charged  by  induction,  and 

sensibly  all  the  tubes  from  one  plate  of  each  condenser  terminate 

upon  the  other. 

Also  F=  Vl  +  F2  +  -  .  -  +  Fn 

Hence 

S  =  f/  V=ql(  V,  +  F2  +  .  .  .  +  FJ 

from  which  (15)  follows  on  cancelling  q. 


CONDENSERS   AND    ELECTROMETERS.  135 

9.  Some  Electrostatic  Methods  of  Comparing  Capacities.  In 
each  of  the  following  methods  the  capacity  of  the  electrometer, 
or  electrometers,  and  connecting  wires  is  supposed  to  be  negli- 
gible in  comparison  with  the  capacities  to  be  compared,  or  else 
to  be  included  with  them. 

(i)  The  capacities  Sl  and  S2  to  be  compared  are  connected  in 
series  with  a  battery  of  electromotive  force  Vt  and  an  electrom- 
eter is  connected  across  the  plates  of  each;  or  an  electrom- 
eter is  connected  across  the  plates  of  one,  for  example  S2,  and 
another,  with  the  battery,  across  the  terminal  plates.  In  the  first 
case  we  have 

S.V^S.V, 
whence 

sj^-rjr,  (16) 

and  in  the  second  case 


whence 

SJS^VjVt-i  (17) 

If  the  leakage  and  absorption  (Chapter  VI.)  of  the  condensers 
are  negligible,  the  two  measurements  may  be  made  in  succession 
with  a  single  electrometer. 

(2)  The  capacities  to  be  compared  are  arranged  to  be  put  in 
multiple  by  a  switch  K.  With  K  open  let  Sv  to  whose  plates 
the  quadrants  of  an  electrometer  are  connected,  be  charged  to  a 
voltage  V,  and  then  connected  in  multiple  with  S2,  when  both 
condensers  will  come  to  voltage  Vr  Then  we  have 


whence 

SJS^r/^-i  (18) 

(3)  In  this  method  the  condensers  whose  capacities  Sl  and  S2 
are  to  be  compared  are  charged  in  multiple  to  the  voltage  FJ 
insulated,  and  then  again  connected  in  multiple,  but  in  such  a 
way  that  the  positive  and  negative  plates  of  I  are  connected  to 


136          ELEMENTS   OF  ELECTROMAGNETIC   THEORY. 

the  negative  and  positive  plates  of  2,  the  final  voltage  being  Vv 
Immediately  after  charging 

ql  =  SL  V    and     q2  =  S2F 
After  the  final  connection  in  multiple 

9t-9t-(Sl-SJV=(Sl  +  S1)ri 
Hence 

S*ISi  =  (V-V2)l(V+V2)  (19) 

It  is  obvious  that  the  above  three  methods  cannot  be  applied 
when  one  or  both  of  the  condensers  are  of  the  guard  ring  type, 
thus  having  more  than  two  conductors.  The  following  method  of 
testing  the  equality  of  the  capacities  of  two  guard  ring  condensers 
was  devised  by  Maxwell.  It  can  also  be  applied  when  only  one, 
or  neither,  of  the  condensers  is  of  the  guard  ring  type.  In 
the  last  case  it  becomes  identical  with  the  last  of  the  preced- 
ing methods,  which  is  an  extension  of  a  method  due  to  Caven- 
dish. 

10.  Maxwell's  Method  of  Testing  the  Equality  of  the  Capacity 
of  a  Guard  Ring  Condenser  and  that  of  any  Other  Condenser.*  Let 
A  be  the  disk,  B  the  guard  ring  and  sheath,  and  C  the  larger 
plate  of  one  of  the  condensers  ;  and  let  A' ',  B' ,  and  C'  be  the 
corresponding  parts  of  the  other.  If  either  condenser,  as  ABC, 
is  of  the  simpler  form  with  only  two  conductors,  we  have  only  to 
suppress  B  and  to  suppose  A  and  C  to  be  the  two  conduc- 
tors, it  being  understood  that  sensibly  all  the  tubes  of  induc- 
tion pass  from  one  plate  to  the  other  when  the  condenser  is 
charged. 

Let  B  be  kept  always  connected  with  C' ,  and  B'  with  C.     Then 

(i)  Let  A  be  connected  with  B,  and  C'  with/,  the  positive 

(for  the  sake  of  definiteness)  terminal  of  a  battery  or  other  source 

of  electrification,  the  other  terminal  of  which  is  connected  to 

*  Maxwell,   Treatise,  §  229. 


CONDENSERS   AND    ELECTROMETERS.  137 

earth  ;  and  let  A'  be  connected  with  B'  and  C  and  with  the 
earth.  The  two  condensers  are  now  charged  oppositely,  so  that 
A  is  positive  and  A'  negative,  and  the  field  of  each  is  sensibly 
confined  to  the  region  between  the  plates. 

(2)  Let  A,  B,  and  C  be  insulated  from/. 

(3)  Let   A    be  insulated   from   B  and   Cf ,  and    A'  from   B' 
and  C. 

(4)  Let  B  and   C'  be  connected  with  Bf  and   C  and  with  the 
earth.     The  charges  on  A  and  Af  remain  unaltered  in  magni- 
tude,  but    are   now    distributed  over   their  whole   surfaces,  the 
fields  no    longer  being    confined  to    the    regions    between    the 
plates. 

(5)  Let  A  be  connected  with  A1 '. 

(6)  Let  A  and  A'  be  connected  with  one  quadr.ant  pair  of  an 
electrometer  E,  the  other  quadrants  of  which  are  earthed.     If  the 
charges  of  A  and  A'  are  equal  in  magnitude,  the  electrification 
wholly  disappears,  since  they  have  opposite  signs,  and  the  elec- 
trometer is  unaffected.      In  this  case  the  fields  connected  with  A 
and  A'  have  the  same  capacities.      Otherwise,  the  electrometer 
will    indicate    positive    or    negative    electrification    according  as 
A    or    A'   has    the    greater  charge   and    therefore    the    greater 
capacity. 

By  making  repeated  tests  and  adjustments,  if  necessary,  the 
capacity  of  a  condenser  constructed  with  movable  conductors  so 
as  to  have  a  variable  capacity,  or  a  condenser  in  the  process  of 
construction,  may  be  made  equal  to  that  of  a  standard  condenser 
of  the  guard  ring  form. 

Other  methods  of  comparing  capacities  are  described  in  Chap- 
ters XII.  and  XIII. 

11.  Some  Methods  of  Extending1  the  Range  of  an  Electrometer.* 
If  the  ratios  of  the  capacities  of  the  condensers  in  the  three  first 
arrangements  described  above  are  known,  the  three  methods  of 
comparing  capacities  may  be  inverted  for  the  measurement  of 

*Cf.  Maxwell,   Treatise,  §220;  Lord  Kelvin,  B,  A.  Report,  1885,  p.  907. 


133 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


high   voltages  with   electrometers  constructed    for  low  voltage 
measurement.     Thus  we  have,  from  (17),  (18)  and  (19), 


(0 

(2) 

(3) 


(20) 


By  measuring  Vv  V  may  be  determined  ;  so  that  by  properly 
choosing  or  adjusting  the  ratio  of  the  capacities  the  range  of  an 
electrometer  may  be  almost  indefinitely  extended. 


CHAPTER  IV. 

GENERAL  ELECTROSTATIC  THEORY.     IDEAL     FIELDS 
CONTAINING  TWO  OR  MORE  DIELECTRICS. 

1.  Generalisation  of  Gauss's  Theorem.*  In  §  23,  Chapter  I., 
this  theorem  was  established  for  a  surface  enclosing  a  single 
homogeneous  isotropic  dielectric,  or  such  a  dielectric  and  con- 
ductors. We  shall  now  show  that  it  holds  for  a  closed  surface 
cutting  any  number  of  such  dielectrics,  or  such  dielectrics  and 
conductors,  To  do  this  it  is  necessary  to  show  only  that  the 
strength  of  a  tube  of  induction  is  not  altered  when  it  passes  from 
one  dielectric  into  another. 

For  this  purpose,  consider  the  electric  field  between  the  plates 
A  and  B  of  a  closed  condenser  containing  two  dielectrics  I  and  2, 
I  being  in  contact  with  A  only,  and  2  in  contact  with  B  only. 
If  the  charge  of  A  is  q,  that  of  B  is  —  q,  and  there  is  no  charge 
upon  the  interface  between  the  dielectrics  I  and  2.  (If  there  are 
charges  due  to  contact,  they  are  equal  and  opposite  at  any  point 
of  the  interface.)  Applying  Gauss's  theorem  to  the  region  I,  we 
find  the  total  strength  of  all  the  tubes  emanating  from  A  to  be  q ; 
and,  likewise,  in  the  region  2,  the  total  strength  of  all  the  tubes 
terminating  upon  B  to  be  q.  Thus  the  total  strength  of  all  the 
tubes  is  unchanged  in  passing  across  the  interface  from  A  to  B. 
And  since  this  result  is  absolutely  independent  of  the  size  or 
shape  of  the  dielectrics,  that  is  of  the  shapes  of  the  tubes,  it 
follows  that  the  strength  of  every  tube  remains  constant  in  cross- 
ing the  interface,  howsoever  the  field  is  divided  up  into  tubes. 

It  may  be  shown  that  the  theorem  is  also  valid  in  the  general 
case  when  the  dielectrics  are  neither  homogeneous  nor  isotropic, 

*See  The  Physical  Review,  September,  1902,  p.  173. 

139 


140 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


but  the  demonstration  lies  outside  the  scope  of  this  work.     In  all 
that  follows  we  shall  assume  the  theorem  to  be  perfectly  general. 

2.  The  (Uncharged)  Interface  Between  Two  Dielectrics.  Laws 
of  Refraction  of  Lines  of  Intensity  and  Displacement.  At  an  un- 
charged interface  S,  Fig.  54,  between  two  dielectrics  I  and  2 
with  permittivities  cl  and  r2,  certain  conditions,  which  we  pro- 
ceed to  find,  must  be  satisfied  by  the  electric  intensity  and  dis- 
placement. 

In  the  first  place,  the  line  integral  of  the  electric  intensity 
around  the  infinitesimal  circuit  adfca,  in  which  ad  and  cf  are  par- 


Fig.  54. 


allel  to  the  interface,  and  ac  and  df  normal  to  the  interface,  is 
zero,  since  the  field  is  static.  This  integral,  that  is,  the  e.m.f. 
around  the  circuit,  is 

El  sin  ei  ad  -f  El  cos  0l  de  +  £2  cos  02ef+E2  sin  Qjc 

-f-  E2  cos  02  cb  -f-  EI  cos  0l  ba  =  o 


FIELDS   WITH    TWO   OR   MORE    DIELECTRICS.          141 

But  de  =  —  ba,  and  ef—  —  cb,  hence  all  the  terms  but  the  first 
and  fourth  cancel,  leaving 

El  sin  6l  ad  -f  E2  sin  62fc  =  o 

or,  since  fc=  —  ad, 

Elsmei  =  E2sme2  (i) 

Thus  the  tangential  component  of  the  intensity  does  not 
change  on  crossing  the  interface. 

Moreover,  E2  lies  in  the  plane  containing  El  and  the  normal  to 
the  interface,  N^2 .  For  if  E2  were  not  in  this  plane,  it  would 
have  a  component  perpendicular  to  this  plane,  while  £l  has  no 
such  component.  Therefore  the  e.m.f.  around  a  circuit  in  a 
plane  perpendicular  to  the  interface  and  lying  partly  in  medium 
I  and  partly  in  medium  2  would  differ  from  zero,  which  is  im- 
possible in  the  static  field. 

Consider  finally  the  electric  flux  outward  across  the  surface  of 
an  elementary  parallelepiped  acfd  with  center  at  o  in  the  inter- 
face ;  two  of  the  faces,  of  breadth  ad  and  height  h  (perpendicular 
to  the  paper)  being  parallel  to  S,  and  the  others,  of  breadth  ac 
and  height  h,  normal  to  5.  This  flux  must  be  zero,  by  Gauss's 
theorem.  Hence 

—  DI  sin  9labh  —  D^  cos  6l  adh  -f  Dl  sin  6l  de  h 

+  D2  sin  02  efh  +  £>2  cos  02  cfh  —  D2  sin  62bch  =  o 

But  since  de  =  ef=  ab  =  be,  and  ad  =  cf,  this  reduces  to 

D2  cos  02  —  Dl  cos  01  =  o 
or 

Dl  cos  6l  =  D2  cos  62    | 

c^  cos  Ol  =  c2E2  cos  62  \ 

Thus  the  normal  component  of  the  displacement  does  not 
change  in  crossing  the  interface. 

From  (i)  and  (2)  we  have  by  division, 

tan  ej  tan  62  =  (DJE^KDJEj)  =  cjc,  (3) 


142  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

In  passing  from  one  dielectric  to  another  lines  of  displacement 
are  therefore  refracted  in  such  a  way  that 

I.  The  incident  and  refracted  lines  are  in  the  same  plane  per- 
pendicular to  the  interface  at  the  point  of  incidence  ;  and  that 

II.  The  ratio  of  the  tangent  of  the  angle  of  incidence  to  the 
tangent  of  the  angle  of  refraction   is  a  constant  for  the  given 
media,  and  equal  to  the  ratio  of  the  two  permittivities. 

Since  tan  6V  and  tan  02  become  infinite  together  when 

*i  =  *,  =  90° 

no  phenomenon  similar  to  total  reflection  in  optics  occurs. 

If  02  is  kept  constant,  and  c2/cl  diminished,  tan  6l  increases. 
In  the  limit  when  c2/cl  =  o,  tan  Ol  =  infinity,  and  6l  =  90°.  In 
this  limiting  case  @2  is  of  course  meaningless,  since  when  c^c^  =  o, 
there  is  no  electric  field  in  medium  2.  As  stated  in  §  14,  L,  no 
substance  has  a  permittivity  less  than  <TO  =  I,  but  for  the  sake  of 
certain  analogies  (VIII.  and  IX.)  the  imaginary  case  of  c^c^ 
=  o  is  here  considered. 

If  while  62  is  kept  constant,  c^c^  is  increased,  6l  increases,  ap- 
proaching o  as  c2  approaches  infinity.  In  this  limit  also  62  is 
meaningless,  and  medium  2  contains  no  electric  field,  as  D 
would  there  be  infinite  if  E  were  greater  than  zero.  Since  in  a 
static  field  the  lines  of  intensity  always  meet  the  surface  of  a  con- 
ductor normally  (6l  =  o)  and  since  there  is  no  electric  field 
within  the  conductor,  a  conductor  behaves  in  a  static  field  like  a 
substance  of  infinite  permittivity.  Since  in  this  case  the  displace- 
ment is  discontinuous  at  the  surface,  the  conductor's  surface  is 
charged.  This  behavior,  however,  is  not  due  to  the  conductor's 
permittivity,  but  to  its  conductivity.  Of  the  permittivity  of  most 
conductors  little  is  known. 

An  experimental  method  of  verifying  (3)  is  described  in  §5, 
VII. 

3.  Fictitious  or  Apparent  Electric  Charges.  The  discontinuity 
in  the  normal  component  of  the  electric  intensity  at  any  point  of 


FIELDS    WITH    TWO  OR  MORE   DIELECTRICS.          143 

the  interface,  viz.,  Ll  cos  Ol  —  E.2  cos  62  (El  and  E2  being  reckoned 
positive  when  directed  from  medium  2  to  medium  i),  is  exactly 
the  same  as  it  would  be  if  c2  were  equal  to  cl  and  there  were  a  dis- 
continuity in  the  normal  component  of  the  displacement  at  the 
point  equal  to  cl  (El  cos  6l  —  E2  cos  02).  This  would  leave  El  and 
E2  everywhere  unaltered,  and  would  leave  Dl  unaltered ;  but 
since  D^  would  now  equal  clE2  instead  of  cJ5,v  as  before,  it  would 
decrease  D2,  and  therefore  all  the  charges  in  medium  2,  in  the 
ratio  cjc2. 

Thus  the  electric  intensity  everywhere  in  the  field  containing 
two  dielectrics  in  contact  (the  interface  being  uncharged)  is  the 
same  as  it  would  be  if  medium  2  were  replaced  by  medium  I, 
if  the  (former)  interface  were  charged  to  a  surface  density 
<r'  =  cl(El  cos  6l  —  E2  cos  02),  and  if  all  the  charges  in  medium  2 
(or  at  the  interfaces,  if  any,  between  medium  2  and  conductors, 
which  could  be  replaced  by  medium  2  (§  28,  I.)  without  altering 
the  field)  were  reduced  in  the  ratio  cjc^  Imagining  these 
changes  made  in  any  case,  we  can  compute  the  intensity  at  any 
point  by  the  direct  application  of  the  law  of  inverse  squares. 
From  the  intensity  and  the  permittivity  at  any  point  the  displace- 
ment can  be  found,  and  from  the  charges  and  the  intensity  the 
mechanical  forces  upon  the  charged  bodies.  This  is  an  extension 
of  the  method  of  §  28,  I.,  which  treats  of  the  case  in  which 
c2  =  infinity,  or  cjc2  =  o,  only.  The  mechanical  force  at  the 
interface  between  the  two  dielectrics  will  be  determined  in  §§  6 
and  9. 

The  quantity     tr'  =  cl(El  cos  Ol  —  E2  cos  02)  (4) 

is  called  the  apparent  or  fictitious  electric  surface  density  at  the 
point  with  respect  to  medium  i.  [In  the  irrational  systems  of 
units,  Chapter  XIV.,  o-'  is  defined  by  the  equation 

471-0-' =  ^(^  cos  0l  —  E2  cos  (92)] 

By  simply  interchanging  the  subscripts  I  and  2  we  could  of 
course  refer  everything  to  the  dielectric  2.  In  all  that  follows, 


144          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

however,  the  medium  designated  as  I  will  be  taken  as  the  stand- 
ard medium,  and  the  apparent  charges,  etc.,  will  be  computed 
with  respect  to  it. 

If  there  are  several  dielectrics  in  the  field,  it  can  be  shown  by 
the  method  used  above  that,  to  reduce  everything  to  medium  I 
for  the  sake  of  computing  the  electric  intensity  by  the  law  of 
inverse  squares,  a  surface  density  must  be  assumed  at  every  point 
of  each  interface  equal  to  cl  x  the  normal  discontinuity  of  E  at  the 
point,  and  the  charges  in  any  medium  of  permittivity  c  must  be 
reduced  in  the  ratio  cjc. 

In  the  same  way,  if  the  permittivity  varies  continuously,  in- 
stead of  suddenly  at  distinct  interfaces,  there  will  be  an  apparent 
volume  density  of  electrification  equal,  at  a  point  where  the  in- 

tensity is  £,  to 

pf  =  ^  div  E  (5) 

(In  the  irrational  systems  of  units  47r//  =  cl  div  E.) 

If  both  volume  density  and  surface  density  of  apparent  elec- 
trification are  present,  we  have 


q'  =     v'dS  +p'dr  (6) 

the  first  integral  being  extended  over  all  fictitiously  charged 
surfaces,  and  the  second  throughout  all  fictitiously  electrified 
volumes. 

Electric  Poles.  The  fictitiously  charged  surfaces  or  volumes, 
that  is,  the  surfaces  or  volumes  where  the  electric  intensity  is 
discontinuous,  are  called  electric  poles.  The  total  apparent  charge 
in  any  region  is  the  strength  of  the  pole  (or  portion  of  a  pole)  in 
that  region,  and  is  equal  to  ^  x  the  outward  flux  of  the  electric  in 
tensity  *  across  a  closed  surface  surrounding  the  pole,  by  (4)  and 
(5).  Another  expression  is  given  below.  The  pole  is  positive 
or  negative  according  as  the  apparent  charge  is  positive  or  nega- 
tive. 

*  The  flux  of  any  vector  across  a  surface  is  the  integral  over  the  surface  of  the  nor- 
mal component  of  the  vector. 


FIELDS   WITH    TWO    OR    MORE    DIELECTRICS.          H5 

4.  Fictitious  Charges  (continued).  Intensity  of  Electrisation. 
Electric  Susceptibility.  (4)  may  be  written 

<r'  =  CI(EI  cos  6l  -  E2  cos  02)  =  (D2  -  c^2)  cos  02         (7) 

The  quantity  Dz  —  c^Ev  the  difference  between  the  actual  dis- 
placement in  medium  2  and  the  displacement  which  would  exist 
there  with  the  same  value  of  intensity  if  c2  were  equal  to  cv  is 
called  the  intensity  of  electrisation  of  medium  2  with  respect  to 
medium  I,  and  is  denoted  by  J.  Thus 

J=D,-c1E2  (8) 

Another  definition  ofj  is  given  in  §12. 

(In  the  irrational  systems  of  units,  Chapter  XIV.,  y  is  defined 
by  the  relation  4?r/=  D2  —  cJE2.) 

When  none  of  the  electrisation  is  intrinsic  (§  I,  VI.),  (8)  may 
be  written  ,. 


/\ 
(9) 

J  is  evidently  a  vector  with  the  same  direction  as  that  of  D2  or 
the  opposite  direction,  according  as  D2  is  greater  or  less  than 
c^E^  ;  or,  when  (9)  is  valid,  according  as  c2  is  greater  or  less 
than  cr 

(9)  may  be  written 

J=(c2-c^  =  lcE2  (10) 

K  —  (c2  —  c^)  is  called  the  electric  susceptibility  of  medium  2 
with  respect  to  medium  I  .  [In  the  irrational  systems  of  units, 

*  =  (C2  ~  'i)/4<l 
(8)  and  (10)  may  be  transformed  into 

D2  =  c2E2  =  /+  c,E2  =  (c,  +  K)E2  (i  i) 

In  terms  of  Jt  the  apparent  surface  density  is 

<r'=/cos02  (12) 

and  the  apparent  charge  upon  a  surface  is 

(13) 


146  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

The  volume  density  of  fictitious  electrification  is 
p'=c1  div  £=  div(Z>— /)  =  div  D— div/=— div/=  conv/  (14) 

since  div  D  =  o. 

The  total  apparent  charge  within  a  volume  r  is 

(15) 

The  total  apparent  charge  in  a  pole  distributed  over  a  surface 
5  and  through  a  volume  r  is 

q'  =  fj  cos  62dS  +  /  convjdr  (16) 

The  total  apparent  charge  within  the  volume  r  and  upon  the 
surface  5  of  a  dielectric  2  completely  surrounded  by  a  homoge- 
neous dielectric  I  is  zero.  This  may  be  proved  by  integrating  the 
first  term  of  (16)  over  the  whole  interface  and  the  second  term 
throughout  the  whole  volume  of  the  dielectric  2.  The  equation 
may  be  written 

q1  =  fD2  cos  62dS  -^  f£2  cos  6.2  dS  -/div  D2dr  +  r1  /div  Ejh 

The  first  and  third  terms  are  evidently  zero,  and  it  will  be  shown 
that  the  second  and  fourth  terms  cancel.  For  div  £2  dr  is  the  ex- 
cess of  the  flux  of  intensity  leaving  the  volume  dr  over  that  enter- 
ing the  volume  dr.  Hence  j  div  E2  dr  throughout  the  volume  r 
is  equal  "to  the  total  excess  of  the  flux  of  intensity  leaving  the 
whole  volume  over  that  entering  the  same  volume  ;  and  this  is 
equal  and  opposite  to  —  cl  j  E.2  cos  62dS,  which  is  the  excess  of 
the  intensity  flux  entering  over  that  leaving  the  whole  volume. 
Thus  the  proposition  is  established. 
A  dielectric  in  which 

p'  =  conv  J  =  c^  div  E  =  o 

is  said  to  possess  solenoidal  electrisation  for  the  reason  that  in 
this  case  all  the  tubes,  or  solenoids,  of  intensity  (E)  or  electrisa- 
tion (J)  run  through  the  dielectric  from  one  pole  face  to  the 
other  without  discontinuity  at  fictitious  charges  between. 


FIELDS   WITH   TWO  OR  MORE   DIELECTRICS.          147 

A  medium  in  which  J  or  D,  together  with  r,  is  constant  is  said 
to  be  uniformly  electrised. 

5.  General  Expression  for  the  Potential  at  a  Point  (  f£cos  6dL 
from  the  point  to  infinity).     When  fictitious  charges  are  present, 
we  must,  to  find  the  potential  at  a  point,  suppose  all  the  true 
charges  reduced  in  the  ratio  cljc,  and  add  to  the  expression  for 
the   potential  due   to   the   true   charges   alone,  (16),  II.,  a  term 

Cdq'  l^rrc^L.     Thus,  in  the  most  general  case, 

V-  I/4WT,  •  (fcjc  •  dqlL+Jdg'jL)^  1/4*  -f^/cL+^'/^L)  (17) 

where  c  is  the  permittivity  at  the  seat  of  the  true  charge  dq.  Or, 
if  we  call  cjc-dq  also  an  apparent  charge,  we  have,  instead  of 

(18) 

6.  The  Integral  Force  Upon  an  Electric  Pole.     The  electric  in- 
tensity Er  at  a  point  P  "  due  to  "  an  electric  pole  of  strength  q' 
is 

E'-fdJlvrc^  (19) 

where  L  is  the  distance  from  the  seat  of  dq'  to  P,  and  the  inte- 
gration is  a  vector  integration,  the  direction  as  well  as  the  mag- 
nitude of  L  being  different  for  each  different  element  dq1  . 

The  force  upon  a  small  body  at  P  with  a  concentrated  true 
charge  q  is 


where  R  is  the  intensity  at  the  seat  of  dqf  due  to  the  charge  q. 
Thus  the  total  force  F  upon  an  electric  pole  is 

F=     Edq'  (20) 


where  R  is  the  intensity  at  the  seat  of  dqf  due  to  the  other  poles 
and  true  charges,  the  integration  being  a  vector  integration. 


148 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


The  traction  per  unit  area  at  any  element  of  an  interface  is  com- 
puted in  §  9. 

The  force  between  two  concentrated  poles  with  apparent 
charges  q'  and  q"  distant  L  apart  in  a  dielectric  of  permittivity  c 
would  be 


F=q'q"JAr>rrcI} 


(21) 


7.  The  Infinite  Parallel  Plate  Condenser  with  Two  or  More  Di- 
electrics in  the  Form  of  Infinite  Plane  Slahs.  The  simplest  pos- 
sible field  involving  two  dielectrics  is  that  of  an  infinite  parallel 
plate  condenser  with  two  dielectrics  I  and  2  of  permittivities  ^ 


Plate  1 


Plate  2 


Fig.  55. 


and  c2  in  the  form  of  infinite  plane  slabs  of  thicknesses  dv  and  d^ 
parallel  to  the  condenser  plates  distant  d  =  dl  -j-  d2  apart  (Fig. 

55). 

Here  the  tubes  of  induction  evidently  run  straight  across  with- 
out change  of  strength  from  one  condenser  plate  to  the  other, 
meeting  both  conductors  normally.  If  Dl  —  D2  =  D  denotes  the 
displacement,  the  intensities  in  media  I  and  2  are 

El  =  DJ^  =  DjCl  and  E2  =  DJc2  =  Djc2 
respectively.      Hence  the  voltage  of  the  condenser  is 

F12  =  E}dl  -f  E2d2  =  D/cl  •  {d  —  \_(c2  —  r^/rj  d2] 

The  capacity  of  a  right  prism  of  the  dielectrics  of  thickness  d 
and  cross-section  A  is 

5  =  A  Dl  ra  =  Acj{d  -  [(,,  -  e^c^  d2]  (22) 

and  the  energy  contained  in  the  prism  is 

W=  ±ADV^  =  lA&lc^{d-  \(cn-  c,)lc2-\d2} 

(23) 


FIELDS  WITH  TWO  OR   MORE    DIELECTRICS.  149 

Thus  the  substitution  of  dielectric  2  for  a  portion  of  dielectric 
I  (cf.  §  51,  Chapter  I.)  decreases  the  energy  if  the  charges  are 
kept  the  same,  and  increases  the  energy  if  the  voltage  is  kept 
the  same,  provided  c2  is  greater  than  cr  If  c2  is  less  than  clt  the 
opposite  is  true. 

If  the  second  dielectric  does  not  touch  either  condenser  plate, 
the  force  upon  either  plate  due  to  the  discontinuity  of  the  dis- 
placement, viz.,  \Ef)  per  unit  area,  is  not  altered  by  its  intro- 
duction when  D  is  kept  the  same;  but  if  Vu  is  kept  the  same, 
the  force  upon  the  area  A  becomes 


F=  \Ac,  VJI  {d  -  [(,,  -  Cl)/cJ  d,Y  (24) 

which  is  greater  or  less  than  when  the  whole  dielectric  had  the 
permittivity  cl  according  as  c2  is  greater  or  less  than  cr 

If  the  dielectric  2  is  in  contact  with  plate  2,  and  the  displace- 
ment D  as  before,  the  force  per  unit  area  upon  plate  I  is,  as  before, 


but  the  force  per  unit  area  upon  plate  2,  due  to  the  discontinuity 
of  the  displacement  at  its  surface,  is 


which  is  less  than  /j  if  c2  is  greater  than  cr  But  the  tension 
along  the  lines  of  intensity  in  medium  I  is  \Ef),  and  in  medium 
2,  ^E2D.  Hence  there  is  a  mechanical  force  upon  dielectric  2 
acting  toward  plate  I  of  magnitude,  per  unit  area, 


-  c^cfr  (25) 

and  this  force  is  transmitted  mechanically  by  the  dielectric  to 
plate  2,  making  the  total  force  per  unit  area  upon  the  plate 
equal  to 


The  apparent  surface  density  at  the  interface  is  uniform  and 

equal  to  .  N  .  , 

cr'  =  D(e   -  Cl)[c,  =/  (26) 


150  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

The  electrisation  is  solenoidal. 

When  the  number  of  dielectrics  is  greater  than  two,  the  inten- 
sity, fictitious  charges,  etc.,  can  easily  be  determined  by  the  same 
method. 

8.  The  Spherically  and  Cylindrically  Radial  Fields.  The  field 
surrounding  a  point  charge  at  the  center  of  any  number  of  con- 
centric spherical  shells  of  different  permittivities,  or  the  field  of  a 
spherical  condenser  with  any  number  of  dielectrics  in  the  form 
of  concentric  spherical  shells,  together  with  the  fictitious  charges, 
etc.,  as  wrell  as  the  cylindrically  radial  field  in  coaxial  cylindrical 
shells  of  dielectric,  can  be  easily  found  by  the  method  of  the 
foregoing  article,  i.  e.t  by  the  direct  application  of  Gauss's 
theorem. 

As  an  example,  suppose  we  have  a  spherical  field  in  three 
dielectrics,  the  charge,  q,  being  in  medium  3,  and  medium  I  sur- 
rounding media  2  and  3  and  extending  to  infinity,  Fig.  56.  The 


Fig.  56. 

displacement  at  any  point  distant  R  from  C  is  ql^cP?,  and 
the  intensity  is  equal  to  the  displacement  divided  by  the  permit- 
tivity at  the  point. 

We  shall  also  find  the  intensity  by  means  of  the  fictitious 
charges.     It  can  be  computed  by  considering  a  charge 


to  exist  at  C  instead  of  q  ;  a  uniformly  distributed  charge 


FIELDS   WITH    TWO   OR   MORE    DIELECTRICS. 


at  the  interface  23  ;  and  a  uniformly  distributed  charge 


at  the  interface  21;  all  in  a  dielectric  of  permittivity  cr  Each 
charge  "produces  "  outside  the  surface  on  which  it  is  distributed 
the  same  effect  as  if  it  were  concentrated  at  the  center  C,  and 
within  the  surface  no  effect  at  all.  Thus  the  intensity  at  a  point 
distant  R  from  C,  when  R  is  greater  than  Rv  is 

E  -  W  +  4*    +  fciOM*^  =  q\V*cJt  (27) 

while  the  intensity  at  a  point  in  medium  2  distant  from  the  cen- 
ter c  by  R,  less  than  R2  and  greater  than  Ry  is 

E  =  fe'  +  OM^i^2  =  qlwcf?  (28) 

9.  The  Mechanical  Force  at  the  Uncharged  Interface  Between 
two  Dielectrics.  In  the  particular  case  considered  in  §  7,  where 
the  lines  of  induction  were  normal  to  the  interface,  there  was 


Fig.  57. 


found  to  be  a  force  at  the  interface  normal  to  it  and  equal,  per 
unit  area,  to  \cJiL*  —  \cf£  measured  in  the  direction  21.  In  this 
article  we  shall  find  the  force  per  unit  area  at  any  point  P  of  the 
interface  in  the  general  case  when  the  lines  of  induction  make 


152  ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

any  angles  (Figs.  54  and  57),  connected  by  the  relations  (i)  and 
(2)  with  the  normal  at  P  to  the  interface. 

In  Fig.  57  let  the  plane  JfF(the  plane  of  the  paper)  coincide 
with  the  plane  containing  El  and  E2  at  the  given  point  of  the 
interface,  the  axis  Z  being  perpendicular  to  XY  through  P  and 
the  plane  of  the  interface  coinciding  with  the  plane  XZ.  Let 
dS  =  dxdz  be  a  rectangular  element  of  area  of  the  interface  at 
the  point,  with  its  sides  parallel  to  X  and  Z  respectfully.  We 
shall  find  the  components  dX  and  dYof  the  force  upon  dS  in  the 
directions  X  and  Y,  parallel  and  perpendicular,  respectively,  to 
the  interface.  It  is  evident,  from  symmetry,  that  the  component 
in  the  direction  Z  is  zero. 

The  force  upon  dS  is  clearly  equal  to  the  force  upon  the  sur- 
face abed  formed  by  drawing  rectangles  a,  b,  c,  d,  all  of  breadth 
dz,  through  the  ends  of  dS,  with  their  planes  parallel  and  per- 
pendicular to  El  and  E2.  The  areas  of  these  rectangles  are 

a  =  dS  cos  0V  b  =  dS  sin  0V  c=  dS  cos  02,  d  =  dS  sin  02 

Let  TI  and  pl  =  Tt  =  \c\E-?  denote  the  tension  and  pressure 
parallel  and  perpendicular,  respectively,  to  the  intensity  in  me- 
dium I  ,  and  T2  and  p2  =  T2  =  \cf£  the  corresponding  quan- 
tities in  medium  2. 

The  force  upon  the  face  a  is  7j  .  a  =  T^S  cos  6V  with  the 
components 

dXu  =  T^S  cos  6l  •  sin  0l     and     dYa  =  T^dS  cos  0l  •  cos  0l 


in  the  positive  directions  of  X  and  K     The  force  upon  the  face 
b  is  pl  •  b  =  pvdS  sin  Ov  with  the  components 


dXb  =  p^dS  sin  0l  •  cos  6l  and  dYb  =  —  p^dS  sin  6l  -  sin  6^ 

in  the  positive  directions  of  X  and  Y.     In  like  manner,  the  com- 
ponents of  the  force  upon  the  faces  c  and  d  are 

dXe  =  -  T2dS  cos  02  •  sin  02,     dYc  =  -  T2dS  cos  62  •  cos  02 
and    dXd  =  -  p2dS  sin  02  •  cos  02,      dYd  =  p.2dS  sin  02  •  sin  02 


FIELDS   WITH    TWO   OR   MORE    DIELECTRICS.          153 

Hence  we  have,  for  the  X  and  Y  components  of  the  total  force 
upon  dSy 

dX=  dXa  +  dXb  +  dXc  -f  dXd  =  \_\c^(2  sin  0l  cos  ^) 

-lr242(2sin02cos 
and 

dYb  +  </Fe  +  </F,  =  [fo^cos2  0L  -  sin2  0 


The  Jf  and  F  components  of  the  force  per  unit  area  upon  the 
interface  at  P  are  therefore 


dXjdS  =  \c^2  sin  0l  cos  0l  -  \c£.*  2  sin  02  cos  (92 

(29) 

=  faEf  sin  2^  —  \c2E*  sin  2^2 
and 

^/F/^/5  =  l^^cos2  0l  -  sin2  0t)  -  l^22(cos2  (92  -  sin2  ^2) 

(30) 
=  ^j^2  cos  20  1  —  \cJEL*  cos  202 

On  multiplying  together  equations  (i)  and  (2),  we  find 

cJE?  sin  20l  =  r2^22  sin  2<92  (3  1) 

Hence 

dXjdS  =  dZjdS  =o  (32) 

and  the  total  force  at  the  interface  is  normal  to  the  surface  and 
equal  to  dYfdS. 

By  making  use  of  (31),  (30)  may  also  be  written 

dYjdS  =  \c&  sin  2(0,  -  00/sin  20,  (33) 

When  cz  is  greater  than  cv  02  —  6l  is  positive  by  (3),  and  dYjdS 
is  positive,  that  is,  directed  toward  medium  i  . 

When  02=01==  o°,  (30)  and  (33)  reduce  to  (25). 

10.  The  Process  of  Changing  the  Dielectric  Within  the  Plates  of 
an  Ordinary  Parallel  Plate  Condenser  is  of  much  interest.  If  the 
plates  have  charges  q  and  —  q,  q  unit  tubes  will  pass  from  one 


154          ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

to  the  other.  When  the  permittivity  of  the  medium  inside  the 
plates  is  the  same  as  that  of  the  medium  outside,  cv  the  tubes 
will  have  some  such  distribution  as  that  indicated  in  Fig.  47. 

If  a  slab  of  a  dielectric  of  greater  permittivity,  c^  is  introduced 
between  the  plates,  the  tubes  will  crowd  into  this  dielectric  until 
(3)  is  satisfied,  leaving  fewer  tubes  in  the  region  outside  than 
before.  That  this  takes  place  follows  from  the  consideration  that 
if  the  induction  between  the  plates  were  not  to  increase,  the 
lateral  pressure  in  that  region,  which  is  proportional  to  D2 fc, 
would  be  insufficient  to  maintain  equilibrium,  c2  being  greater 
than  cv  and  equilibrium  having  existed  when  c^  was  equal  to 
clt  or  when  the  dielectrics  outside  and  inside  were  the  same. 
Since  the  dielectric  outside  is  unaltered,  we  can  compare  the 
voltages  of  the  condenser  in  the  two  cases  conveniently,  if  we 
measure  them  along  the  same  path  outside  before  and  after  the 
introduction  of  the  slab.  Since  the  field  in  the  external  region 
is  weaker  after  the  introduction  than  before,  the  voltage  is  seen 
to  be  less.  If  ^  is  greater  than  c2,  the  effects  are  of  course 
opposite. 

To  look  at  the  matter  in  another  way,  the  tubes  connecting 
the  outside  surfaces  and  those  connecting  the  inside  surfaces  may 
be  regarded  as  the  tubes  of  two  condensers  connected  in  parallel. 
If  the  capacity  of  either  is  increased  by  increasing  the  permittivity 
of  its  dielectric,  the  tubes  will  crowd  into  that  one,  and  the  com- 
mon voltage  of  both  will  be  reduced.  If  the  distribution  of  the 
tubes  remained  unaltered,  the  voltage  between  the  two  plates 
would  be  greater  along  a  line  not  passing  through  the  slab  than 
along  a  line  passing  through  the  slab. 

During  the  lateral  introduction  of  the  slab  into  the  region 
between  the  condenser  plates,  the  tubes  crowding  into  it  exert  a 
pull  upon  it,  by  §  9,  which  continues  until  it  is  symmetrically 
situated  with  respect  to  the  plates,  when  the  pulls  urging  it  in 
all  directions  balance.  If  during  the  change  the  charges  are  kept 
constant,  the  energy  decreases,  since  the  voltage  decreases  ;  if 
the  voltage  is  kept  constant  the  charges  increase  and  the  energy 


FIELDS    WITH    TWO  OR    MORE    DIELECTRICS. 


155 


ncreases. 


By  computing  the  space  rate  of  this  increase  or 
decrease  of  the  energy,  the  force  acting  upon  the  slab  may  be 
found  by  the  method  of  §  55,  Chapter  I.  This  computation  is 
made  in  §  4,  Chapter  VII.  The  force  could  not,  in  general,  be 
determined  without  very  great  difficulty  by  the  method  of  §  9. 

11.  Field  Surrounding  a  Concentrated  Charge  Situated  in  One 
of  Two  Infinite  Dielectrics  Separated  by  a  Plane  Interface.  Let 
the  charge  q  be  concentrated  at  the  point  A,  Fig.  58,  in  the  me- 


D 
Fig.  58. 

dium  of  permittivity  cl  distant  d  from  the  interface  separating  the 
medium  of  permittivity  ci  from  that  of  permittivity  c2. 

By  §  48,  I.,  there  is  only  one  field  which  can  satisfy  the  given 
conditions.  The  given  equipotential  in  this  case  is  the  infinite 
sphere  at  zero  potential  with  A  as  center.  To  find  the  field  by 
means  of  the  law  of  inverse  squares,  we  must  reduce  the  problem 
to  one  with  a  single  dielectric,  until  the  displacement,  or  else  the 
intensity,  everywhere  is  found.  Then  the  unknown  one  of  the 
two  can  be  found  in  each  dielectric  from  the  relation  D  =  cE. 
We  shall  combine  the  method  of  images  with  the  method  of  §  3. 

Guided  by  the  result  of  §  15,  II.,  which  solves  the  problem 
when  £~2  =  infinity,  and  by  what  we  have  learned  of  the  refraction 
of  lines  of  displacement,  the  simplest  rational  assumption  we  can 


156          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

make  is  that  to  the  left  of  CD  the  displacement  is  such  as  would 
accompany  a  charge  q  at  A  and  a  charge  ql  at  B,  both  in  a  di- 
electric of  permittivity  cl  ;  and  that  the  displacement  to  the  right 
of  CD  is  such  as  would  emanate  radially  in  the  medium  I  from 
a  charge  q2  at  A.  If  <?l  and  q2  can  be  given  such  values  as  to 
satisfy  (i)  and  (2),  at  every  point  of  CD,  the  assumption  will  be 
justified  and  the  problem  solved. 

Choosing  OB  and  OC  as  positive  directions,  we  find,  for  the 
normal  displacement  at  P  on  the  left  side  of  CD, 


*  =  (q  - 
and  for  that  on  the  right  side 


(2)  Will  be  satisfied  if  these  are  equal.     Hence,  if  the  prob- 
lem can  be  solved  by  this  method, 


The  intensity  parallel  to  CD  is,  on  the  left, 
and  on  the  right, 
Hence,  to  satisfy  (i),  we  must  have 


Equations  (a)  and  (S)  are  both  satisfied  by  the  values 
and  ft  =  -*(',  -'i)/(',  +  ' 


so   that   the   above  assumptions   are  justified  and  the  problem 
solved. 

When  c2  fa  =  f  ,  ql  =  —  q  /4  and  q2  =  t>q  /4.  When  <r2  fa  =  f  , 
ql  =  qJ4  and  ^2=3^/8.  The  plane  diagrams  of  the  field  for 
these  two  cases  are  easily  constructed  from  Figs.  23  and  24,  or 


FIELDS    WITH    TWO    OR    MORE    DIELECTRICS. 


157 


from  the  above  charges  directly,  by  Maxwell's  method,  §  14,  II. 
The  diagrams  of  the  tubes  of  displacement  for  these  two  cases  are 
given  in  the  upper  and  lower  halves,  respectively,  of  Fig.  59. 
The  dotted  lines  on  the  right  of  the  vertical  line  and  the  full 


Fig.  59. 


lines  on  the  left  are  the  lines  of  displacement  of  Figs.  23  and  24. 
The  force  upon  the  charged  body  at  A,  or  the  force  between 
the  charged  body  at  A  and  the  dielectric  2,  is 


F=  qq,  U-Kc^Ldy  =  -  f(c2  -  cj  llbTrd2^  +  Cl)       (35) 

If  c2  is  greater  than  cv  the  force  is  one  of  attraction,  the  tubes 
being  concentrated  on  the  side  of  A  toward  medium  2  ;  but  if  c2 
is  less  than  cv  the  force  is  one  of  repulsion,  the  tubes  being  now 
concentrated  on  the  opposite  side.  When  c2  =  infinity  (35)  re- 
duces to  (41),  II.  In  the  first  case  the  apparent  charge  upon 
the  interface  is  negative,  in  the  second  positive,  and  in  the  last  the 
charge  is  real  and  negative,  q  being  supposed  positive. 


158  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

The  total  force  upon  the  interface  can  be  obtained  also  by  the 
method  of  §  6. 

Thus  the  apparent  surface  density  at  P  is 

<r'  =  -  qd(ct  -  fl)  /2TT(c2  +  c^  (36) 

The  apparent  charge  dq'  upon  a  zone  in  the  interface  of  radius 
OP  and  width  d(OP)  is 


dqf  =  (7f27rOP  -d(OP)  =  ffr2irx 

The  normal  intensity  at  the  zone  due  to  the  charge  q  is 


Hence  the  total  force  between  the  interface  and  the  charged 
body  at  A  is 


F=  -  fd\ct  -  cflvrcfa  +  O  f 

Jd 


as  in  (3  5). 

The  total  apparent  charge  upon  the  interface  is 


'  =    I     <T'2irxdx=  —  qd(c2  —  c^  /(c2  +  ^)  I     dx  l# 

Jd  Jd  (37) 


12.  Dielectric  Sphere  in  a  Uniform  Field  of  Different  Permit- 
tivity. Let  a  sphere  of  permittivity  c2  be  introduced  into  an 
infinite  medium  of  permittivity  cl  supporting  (before  the  introduc- 
tion of  the  sphere)  a  uniform  electric  displacement  D. 

If  c2  =  cv  the  tubes  will  remain  everywhere  unaltered. 

If  c2  is  greater  than  cv  the  tubes  will  bend,  crowding  into  the 
sphere,  thus  making  D2  greater  than  D,  until  the  condition  ex- 
pressed in  (3)  is  satisfied.  (If  Z>2  were  to  remain  equal  to  Dt  the 
lateral  pressure  ^D2jc2  across  the  tubes  within  the  sphere  would 
be  less  than  JZ?2/^,  the  lateral  pressure  without,  and  equilibrium 
could  not  exist.  Also,  the  voltage  between  two  equipotentials 


FIELDS   WITH    TWO    OR    MORE    DIELECTRICS. 


159 


would  be  greater  along  a  line  not  traversing  the  sphere  than 
along  a  line  passing  through  the  sphere.) 

If  c2  is  less  than  cv  D2  is  less  than  D,  tubes  crowding  out  of 
the  sphere  until  (3)  is  satisfied. 

We  proceed  to  the  exact  determination  of  the  electric  field 
within  and  without  the  sphere.  In  accordance  with  §  48,  L, 
there  is  but  a  single  field  satisfying  the  conditions  of  the  problem. 

With  respect  to  the  field  within  the  sphere,  we  shall  make  the 
simplest  possible  rational  assumption,  viz.,  that  the  displacement 
D2  is  uniform  and  in  the  same  direction  as  the  original  external 
displacement  D.  We  shall  further  assume  that  the  effect  of  the 


Fig.  60. 

sphere  on  the  (originally)  uniform  field  is  the  same,  in  the  region 
outside  it,  as  that  of  a  doublet  of  moment  M  placed  in  the 
original  dielectric  at  the  point  occupied  by  the  center  of  the 
sphere  with  its  axis  parallel  to  D.  The  probable  correctness  of 
this  assumption  follows  from  the  fact  that  conductors  and  dielec- 
trics produce  on  static  fields  into  which  they  are  introduced 
effects  differing  only  in  degree  ;  and  the  fact  that  the  effect  of  a 
conducting  sphere  on  a  uniform  field  can  be  represented,  in  the 
region  outside  the  sphere,  by  a  doublet  at  its  center. 

An  attempt  at  a  solution  based  on  these  assumptions  will 
obviously  satisfy  all  the  electrical  conditions  except  (i)  and  (2). 
If  in  addition  D.2  and  J/can  be  so  chosen  as  to  satisfy  these  con- 
ditions also,  the  problem  will  be  solved. 


i6o 


ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 


To  see  whether  the  assumptions  made  above  will  satisfy  (i) 
and  (2),  the  radial  components  of  the  resultant  internal  and  ex- 
ternal displacements  and  the  tangential  components  of  the  re- 
sultant internal  and  external  intensities  at  the  surface  of  the 
sphere  must  be  determined.  The  radial  and  tangential  displace- 
ments at  any  point,  outside  the  sphere,  whose  coordinates  are  R 
and  6,  in  the  notation  of  §§  25  and  26,  II.,  will  first  be  found. 

For  the  radial  component  due  to  the  doublet  (94),  II.,  gives 
2M  cos  0/4.7rR3,  to  which  must  be  added  the  component  D  cos  6 
due  to  the  uniform  field.  For  the  tangential  component  due  to 
the  doublet  (95),  II.,  gives  M  sin  Oj^rrR^,  to  which  —  D  sin  6, 
due  to  the  uniform  field,  must  be  added.  For  the  total  radial 
displacement  outside  the  sphere  we  have,  therefore, 


D  =  2Mcos 


+  D  cos  0 


and  for  the  total  tangential  displacement, 

Dt  =  M  sin  6/4>irlP  -  D  sin  0 


(38) 


(39) 


Fig.  61. 


At  any  point  inside  the  sphere,  with  coordinates  R  and  0,  the 
radial  and  tangential  components  of  the  displacement  are  D2  cos  0 
and  —  D2  sin  0,  respectively,  independently  of  the  value  of  R 
(less  than  the  radius  of  the  sphere). 


FIELDS   WITH    TWO    OR    MORE    DIELECTRICS.          l6l 

Let  the  radius  of  the  sphere  be  denoted  by  a.     Then,  to  satisfy 
(i)  and  (2),  we  must  have,  when  R  =  a, 

2M  cos  0/47rtf3  +  D  cos  0  =  D2  cos  6,     or    M/27ra*  -f  D  =  D2 
and  M  sin  O/c^Tra*  —  Djcl  •  sin  0  =  —  D2  sin  6/c2, 

or 

The  solution  of  these  equations  gives 

M=  ^Jra\c2  -  cjDj(c2  +  2^)  =  4™3fo  -  '0  A/3',      (40) 

and  A  =  y2Dj(c2  +  2^) 

(41) 
whence  .£  =       EI(c  +  2r 


The  assumptions  made  above  are  therefore  justified,  and  the 
problem  is  solved.  The  uniform  field  within  the  sphere  is  given 
by  (41),  and  the  external  field  by  (38)  and  (39)  on  substituting 
for  Mits  value  from  (40).  This  substitution  gives 

Dr  =  [2a\c2  -  c})jR\c2  +  2^)  -f  I  ]  D  cos  6  (42) 

which  becomes,  when  R  =  a, 

Dn=yJ)CQ*OI(ct+2cd  (43) 

also  Dt  =  [J(c2  -  c^jR\c2  +  2^)  -  i  ]  D  sin  6  (44) 

which  becomes,  when  R  =  a, 

A«  =  -  3^  sin  0/^+2^)  (45) 

When  c2  is  greater  than  cv  D2  is  greater  than  D,  E2  is  less  than 
E,  and  M  is  positive.  That  is,  the  doublet  is  turned  with  its 
positive  end  in  the  direction  of  the  field.  When  c2  is  less  than  cv 
D2  is  less  than  D,  E2  is  greater  than  E,  and  M  is  negative,  or  the 
doublet  is  turned  so  as  to  oppose  the  field.  When  c2  =  infinity, 
(41)  reduces  to  D2  =  ^D,  and  E2  =  o  ;  and  (38),  (39),  and  (40) 
to  the  equations  of  §§  25-27,  II. 

The  plane  diagrams  of  the  tubes  of  displacement,  drawn  by  the 
method  of  §  14,  II.,  for  c2/cl  =  o,  3,  and  infinity,  respectively,  are 


1 62  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

given  in  Figs.  60  (the  lines  inside  the  circle,  Fig.  32,  formed  by 
superposing  the  uniform  field  on  the  field  of  the  doublet,  being 
here  annulled),  61,  and  62  (from  Webser's  Theory  of  Electricity 
and  Magnetism,  §  1 94). 

The  infinite  plane  passing  through  the  equator  of  the  sphere 
is  an  equipotential  surface  (at  zero  potential).  Hence  if  this  sur- 
face is  made  conducting  we  shall  have  on  each  side  half  the  field 
just  considered  terminated  by  this  conducting  sheet.  Thus  we 
have  solved  the  problem  of  finding  the  field  terminated  by  an 


Fig.  62 

infinite  plane  conducting  surface  with  a  hemispherical  boss  upon 
it  of  permittivity  c2  differing  from  that  of  the  dielectric  occupying 
the  rest  of  the  field  (^). 

The  electric  surface  density  at  any  point  of  this  plane  distant 
R  from  the  center  of  the  hemisphere  is 


when  R  is  less  than  #,  and 

90°)  =  ±  \_a\c,  -  Cl}IR\c,  +  2,,)  -  i]/>  (47) 


when  R  is  greater  than  a. 

The  intensity  of  electrisation  of  the  sphere  is  uniform  and  equal 

to  /  =  D,  fa  -  OM]  =  3D(c,  -  O/k  +  2^)  (43) 


FIELDS   WITH    TWO   OR   MORE    DIELECTRICS.          163 

(40)  may  now  be  written,  in  terms  of  Jt 

M=  f  TT</  (49) 

so  that  the  intensity  of  electrisation  of  the  sphere  might  be  defined 
as  its  electric  moment  per  unit  volume,  the  electric  moment  denot- 
ing the  moment  of  the  doublet  producing  the  same  effect  on  the 
external  field  as  that  of  the  sphere. 

The  difference  between  the  actual  intensity  E2  in  the  sphere 
and  the  original  intensity  E  of  the  uniform  field  is  called  the  self- 
deelectrising  force  or  intensity  in  the  sphere  due  to  its  poles  or 
apparent  charges,  and  will  be  denoted  by  E'  .  Thus 

The  apparent  electric  surface  density  at  a  point  whose  coordi- 
nates are  a  and  6  is 

•r'  =  />,[(',-',)/',]  «>S0-/  cos* 
=  3  [fo  -',)/(',  +  2*1)]/>  cos* 

The  total  apparent  charge  upon  one  half  of  the  sphere  between 
a  pole  and  the  equator  is 


q<  =  ±  ™y  =  ±  TO2[(<-2  -  c^l^Dt  (52) 

13.  Infinite  Dielectric  or  Conducting  Cylinder  in  a  Uniform 
Field.  Making  use  of  §  20,  II.,  we  can  obtain,  by  the  method  of 
the  preceding  article,  the  electric  field  in  and  about  an  infinite 
circular  cylindrical  dielectric  of  permittivity  c2  immersed  in  the 
uniform  field  of  an  infinite  medium  whose  permittivity  is  cr  For, 
as  will  be  seen,  it  is  possible  to  satisfy  all  the  conditions  by 
assuming  the  displacement  external  to  the  cylinder  to  be  the 
resultant  of  the  original  uniform  displacement  and  the  displace- 
ment of  a  line  doublet  of  moment  M,  suitably  chosen,  at  its  axis, 
all  in  the  original  dielectric,  and  the  internal  displacement  to  be 
uniform  and  parallel  to  the  original  displacement.  If  the  radius 
of  the  cylinder  is  a,  and  if  D  and  D2  denote  the  original  uniform 


1  64  ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

displacement  and  the  actual  displacement  within  the  cylinder, 
respectively,  the  conditions  (i)  and  (2)  to  be  satisfied  at  the 
interface  are  obviously 

D  cos  6  +  Mcos  6/27ra2  =  D2  cos  6 
and 

—  D  sin  e/^  +  ^sin  e/c^Tra2  =  —  D2  sin  0/c2 

from  which  we  obtain 

A  =  2^M  +  O  (53) 

and    M=  7ra*[(c2  -  c^jc^D,  =  ^a\(c,  -  ,,)/(,,  +  ,,)]/>    (54) 

Hence  outside  the  cylinder,  at  a  distance  R  from  its  axis,  the 
radial  component  of  the  total  displacement  is 

Dr  =  Mcos  6/27rR2  +  D  cos  0 

=  [a2/R2-(c2  -  c^(c%  +  cj  +  i\D  cos  B  (55) 

and  the  tangential  component  is 

Dt  =  [*!&•(*,  -  c,)l(ct  +  ,,)  -  i]Z>  sin  0  (56) 

while  within  the  cylinder  the  displacement  is  uniform  and  given 

by  (53). 

The  apparent  electric  surface  density  is 

»'  =  \.(^  -  «0/'J  A  cos  *  -/  cos  ^  (57) 

The  total  positive  or  negative  apparent  charge  on  half  of  unit 
length  of  the  cylinder  is 

ql=Jx2ax  i  =2«/  (58) 

The  self-deelectrising  force  of  the  apparent  charges  is 

E'  =  Dt  lct  -  Die,  =  -  (ct  -  Cl)  1  2cft  •  D2 

(59) 


14,  Dielectric  Spherical  Shell  in  a  Uniform  Field.  Electric 
Screen.  If  instead  of  the  solid  sphere  of  §  12,  we  have  a  spherical 
shell  of  permittivity  cv  with  inner  and  outer  radii  b  and  a  respec- 


FIELDS    WITH    TWO  OR    MORE    DIELECTRICS.          165 

lively,  surrounding  and  surrounded  by  a  medium  of  permittivity 
cl  supporting  an  (originally)  uniform  field  with'  displacement  D,  we 
can  find  the  field  by  an  extension  of  the  method  used  in  the  two 
preceding  articles. 

Guided  by  the  results  obtained  for  the  solid  sphere,  we  shall 
assume  (  I  )  that  within  the  inner  surface  of  the  shell  the  dis- 
placement, Dz  is  uniform  and  parallel  to  that  of  the  original  field; 
(2)  that  within  the  shell  the  displacement  is  the  vector  sum  of  a 
uniform  displacement  D2  parallel  to  D  and  the  displacement  due 
to  a  point  doublet  of  moment  Mb  at  the  center  of  the  spheres  ; 
and  (3)  that  the  displacement  outside  the  shell  is  the  vector  sum 
of  the  uniform  displacement  D,the  displacement  due  to  the  doublet 
of  moment  Mb,  and  the  displacement  due  to  a  second  doublet  of 
moment  Ma,  also  placed  at  the  center  of  the  spheres,  the  axes  of 
both  doublets  being  parallel  to  D.  It  will  now  be  shown  that 
these  assumptions  satisfy  (i)  and  (2). 

At  the  outer  interface  the  conditions  to  be  satisfied  by  the 
normal  displacement  and  tangential  intensity  are,  respectively, 

(D  +  2Ma/47ra*  +  2Mb/fir<t)  cos  d  =  (D2+  2^/4™*)  cos  6 
and 
(-  D/CI  +  MJc^TTC?  +  MJc^TraP)  sin  B 

sin  0 


At  the  inner  interface  the  conditions  are 

(D2  +  2MJ47T&)  cos  6  =  D3  cos  0 
and  (-  £>Jc2  +  MJc^irP)  sin  0  =  -  DJcl  .  sin  6 

Cos  6  and  sin  6  will  divide  out,  and  the  equations  are  satisfied  by 
the  following  values  of  the  assumed  moments  and  displacements  : 

A  =  9V.0/IW,  +  2fc  -  rf(i  -  #/«•)]  (60) 

A  =[(2^  +  ^/3',]  A  (60 

Ma  =  27ra3(JD2  -  D)  (62) 

and  Mt  -  -  4^/3  •  [(,,  -<-,)/<-,]  Dt  (63) 

The  relation  betwen  D^D  and  bja  is  given  in  the  accompany- 
ing table  (Table  I.)  for  the  cases  in  which  cjcl  =  100  and  1000. 


1 66 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


These  excessive  values  of  c2jcl  do  not  occur  in  electrostatics,  but 
are  assumed  here  for  the  sake  of  the  much  more  important  mag- 
netic analogue,  §  23,  XL  The  greater  the  ratio  of  c2  to  cl  and 
the  smaller  the  ratio  of  b  to  a,  the  less  is  D3  in  comparison  with 
D,  that  is,  the  shell  forms  a  more  effective  screen  from  electrical 
influences  for  the  region  within  it.  When  c2/cl  =  infinity,  that  is, 
when  the  shell  is  conducting,  DJD  =  o  for  all  values  of  bja,  and 
the  shell  is  a  perfect  electrical  screen  (when  the  field  is  static). 

15.  Dielectric  Cylindrical  Shell  in  a  Uniform  Field.      Electric 
Screen.     The  field  of  an  infinitely  long  circular  cylindrical  shell, 


Fig.  63. 


of  permittivity  c2  and  with  inner  and  outer  radii  b  and  a,  when 
immersed  in  an  infinite  medium  of  permittivity  ^  supporting  an 
(originally)  uniform  displacement  D  can  be  obtained  in  exactly 
the  same  way,  by  making  use  of  line  doublets  (§  20,  II.)  instead 
of  point  doublets. 

Let  D,  D2,  D^  MaJ  and  Mb  have  the  same  meanings  as  in  §  14, 
except  that  cylinder  and  cylindrical  must  be  substituted  for 
sphere  and  spherical  and  line  doublet  for  point  doublet.  Then 
we  have,  to  satisfy  (i)  and  (2),  at  the  outer  interface, 

(D  -f  Mj27ra2  +  MJ27ra2)  cos  0  =  (Da  +  Mb/27ra2)  cos  6 
and 


FIELDS   WITH    TWO   OR    MORE    DIELECTRICS.          l6/ 

sin<9 


and  at  the  inner  interface, 

(£>2  +  Ma/27r&2)  cos  6  =  Z>3  cos  (9 


and 


(-  D2/c2  +  Mb  /c227r&2)  sin  0  =  — 


•  sin  0 


Solving  these  equations,  we  obtain,  as  the  only  solution  of 
the  problem, 

-;;;  ,      /7,-4/v,  /[(',  +  ',)•  -*/«••(',  -'in         (64) 
A-fo  +  'i)/2vA  (65) 

Ma=27ra2(£2-D)  (66) 

and      ^  =  -^[(,,-0  /,,]/>,  :_";•:::..:-::.".-;    (67) 

The  relation  between  Z>3  /  D  and  £/#  is  given  in  the  accompa- 
nying table  (Table  I.)  for  the  cases  in  which  c2jcl  =  100  and 
1000.  When  c^c^  =  infinity,  or  when  the  shell  is  a  conductor, 
D^  I  D  =  o  for  all  values  of  b  /a.  The  cylindrical  shell,  like  the 
spherical  shell  of  §  14,  forms  an  electrical  screen,  the  remarks  at 
the  close  of  §  14  applying  equally  well  to  both  forms. 

The  plane  diagram  of  the  tubes  of  displacement  when  c2jc^  =  10 
and  b  la  =  -|  is  given  in  Fig.  63  (from  Webster's  Theory  of  Elec- 
tricity and  Magnetism,  §  198). 

TABLE  I. 
SCREENING  EFFECT  OF  SPHERICAL*  AND  CYLINDRICAL  DIELECTRIC  SHELLS. 


b\a 

DZ\D  when  <r2/f  t  =  100. 

DZ\D  when   fj/^!  =  1000. 

Spher.  Shell. 

Cyl.  Shell. 

Spher.  Shell. 

Cyl. 

Shell. 

0.0 

ft 

•fa 

irk 

rfr 

O.I 

TS 

A 

^^3" 

7~ 

T 

0.2 

3jC 

2T                                    'STT 

T 

o-3 

^V 

^"3"                                    fT^" 

y 

f 

0.4 

rt 

¥ 

^ 

T 

°-5 

A 

_l 

T' 

0.6 

iV 

T7 

T77 

d 

0.7 

IT 

JL 

T^ 

-J 

0.8 

T2 

I 

rib 

s 

0.9 

4 

'sV 

' 

0.99 

| 

1 

T\ 

J 

I.O                               I.O 

I.O 

I.O 

O 

*  The  data  for  the  spherical  shell  are  taken  from  J.  J.  Thomson's  Elements  of 
the  Mathematical  Theory  of  Electricity  and  Magnetism,  §  1 6 1. 


CHAPTER  V. 

REVERSIBLE    THERMAL    EFFECT    DURING    ELECTRISATION. 
ELECTROSTRICTION. 

1.  Reversible  Thermal  Effect  During  Electrisation.  Let  a  con- 
denser be  carried  through  a  reversible  cyclic  process  as  follows, 
the  external  pressure  upon  the  dielectric  (e.  g.t  the  atmospheric 
pressure)  being  kept  constant  : 

(1)  The  voltage  V  being  kept  constant,  let  the  condenser  be 
heated  from  the  absolute  temperature  /  to  the  absolute  tempera- 
ture t  +  dt.     If  p,  s,  and  r  denote  the  density,  specific  heat,  and 
volume,  respectively,  of  the  dielectric,  the  heat  absorbed  by  the 
condenser  during  this  process  is  H  =  pST  dt. 

(2)  At  the  temperature  t  -f  dt,  at  which  the  capacity  of  the 
condenser  is  S  -\-dSldtdt,  let  the  voltage  be  increased  by  dV. 
The  energy  of  the  condenser  increases  by  \(S  +  dSjdt  dt)d(  V2) 
=  (S+  dSjdtdt)VdV. 

(3)  Let  the  condenser  be  cooled  to  the  original  temperature  / 
while  the  voltage  remains  constant  (V  +  dV). 

(4)  Let  the  voltage  be  reduced  to  its  original  value  V,  the  con- 
denser thus  losing  an  amount  of  energy  equal  to 


The  condenser  is  now  in  its  original  condition. 

The  total  work  done  upon  the  condenser  (exclusive  of  work 
done  in  heating)  during  the  complete  cycle  is 

dW=  (S  +  dSjdt  dt)  VdV—  S  VdV=  VdSjdtdt  •  dV 

The  quantity  of  heat  given  to  the  condenser  (exclusive  of  that 
given  out)  is 


168 


REVERSIBLE    THERMAL   EFFECT.  169 

Hence,  by  the  second  law  of  thermodynamics,  viz.,  Hjt  =  —  ' 

dWjdt,  we  have 

ps-rdtjt  =  -  VdVdSjdt 

or  dtjdV=  -  tVips-r  •  dSjdt  (i) 

If  therefore  we  assume  that  dS  j  dtt  s,  etc.,  are  independent  of 
V,  which  is  certainly  near  the  truth,  the  total  reversible  temper- 
ature change  when  the  condenser  is  charged  from  V=  o  to  V= 
Fis 


f 

Jo 


(2) 


If  the  dielectric  is  homogeneous  and  isotropic,  we  have 
=  \lcL-  d(cL]\dt=  ijc-dcjdt+  ijL-  dLjdt  =  kt 


where  L  is  the  length  of  any  line  drawn  in  the  dielectric  (S  and 
dS  being  proportional  by  the  same  factor  to  the  product  of  the 
permittivity  and  such  a  length),  and  kt  and  a  are  written  for  ijc- 
dcjdt,  the  coefficient  of  increase  of  c  with  temperature,  and  i/L- 
dLjdt,  the  coefficient  of  linear  expansion  with  temperature,  re- 
spectively. By  substituting  (3)  in  (2),  we  obtain 


tolt=-(kt  +  a)lfKT-$SV*  (4) 

If  the  field  of  the  condenser  is  uniform,  (4)  becomes 

A///  =  -  (kt  +  a)/ps  -  \cE*  =  -  (kt  +  a)  U/ps  (5) 


Since,  moreover,  any  electric  field  is  uniform  in  its  infinitesimal 
parts,  (5)  is  perfectly  general. 

In  all  the  above  the  effects  of  conduction,  radiation,  etc.,  are  neg- 
lected, and  no  intrinsic  electrisation  (VI.)  is  supposed  to  be  present. 

For  all  solids  yet  investigated  (kt  -\-  a)  is  positive.  Hence  a 
condenser  with  such  a  dielectric  is  cooled  by  charging  and  heated 
by  discharging.  For  nearly  all  liquids  (kt  -f  a)  is  negative.  Ac- 
cording to  experiments  by  W.  Cassie  (Phil.  Trans.,  A,  1890)  k 


1 70         ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

«(or  kt  -f  a)  at  50°  C.  is  about  —  0.006  for  glycerine  and  -f  0.0004 
for  mica.  For  recent  literature  see  Ann.  der  Physik,  Vol.  10, 
p.  748,  1903. 

[Analogous  magnetic  quantities  being  substituted  in  (5)  for 
the  electric  quantities  there  occurring,  the  equation  is  valid  for 
the  magnetic  case  (see  §  23,  XL).] 

2.  Electrostriction.  Change  in  Volume  of  Dielectric  when 
Electrised.  When  a  condenser  is  charged  at  constant  tempera- 
ture its  dielectric,  or  dielectrics,  would  be  expected,  in  general, 
to  suffer  changes  in  volume  and  changes  in  linear  dimensions. 
These  phenomena,  as  yet  largely  hypothetical,  are  included  under 
the  general  head  of  clcctrostriction.  The  alterations  in  volume, 
etc.,  can  be  deduced  from  the  principles  already  developed,  in 
connection  with  the  principle  of  the  conservation  of  energy. 


Fig.  64. 

In  all  that  follows  it  will  be  assumed  that  the  condenser  plates 
are  always  in  contact  with  the  dielectric  and  that  they  follow 
accurately  without  appreciable  elastic  reaction  the  motion  of  its 
surfaces,  as,  for  example,  coats  of  gold  leaf  or  tin  foil.  Complete 
absence  of  intrinsic  displacement  will  also  be  assumed. 

First  we  shall  find  the  change  in  volume.  Consider  a  con- 
denser ABC,  Fig.  64,  whose  dielectric  C  occupies  the  volume  r 
and  possesses  the  permittance  vS  when  charged  to  the  voltage  V 
and  subjected  to  the  uniform  pressure  p  (which  may  have  any 
value,  including  o)  over  its  surfaces. 


ELECTROSTRICTION.  1  7  1 

(1)  While  Fis  kept  constant,  let  the  volume  be  increased  by 
dr.     The  energy  of  the  condenser  will  increase  by  —  pdr.     The 
increase  of  volume  will,  in  general,  be  accompanied  by  an  increase 
dS  in  the  permittance.     Now  let  the  volume  (r  4-  dr)  be  kept 
constant  while  the  voltage  is  increased  by  dV.     The  energy  in- 
creases by  \  (S  +  <tS)  d(  F2)  =  (S  +  dS)  VdV.     The  total  increase 
in  energy  during  the  process  is 

dW^  -  pdr  +  (S  +  dS)VdV 

(2)  Let  us  start  with  the  condenser  in  the  same  condition   as 
at  the  beginning  of  (i)  and  bring  it  to  the  same  final  state   by  a 
slightly  different  process.     While  the  volume  remains  constant 
(T),  let  the  voltage  be  increased  by  dV.     This  will  increase  the 
energy  of  the  condenser  by  SVdV,  and,  in  general,  the  pressure 
by  an  amount  dp.     Now  let  the  voltage  (  F+  dV)  be  kept  con- 
stant while  the  volume  is  increased  by  dr.     The  energy  will  in- 
crease by  —  (J>  -f  dp}dr.     The  total  increase  in  the  condenser's 

energy  is  thus 

dWz  =  SVdV-  (p  +  dp)  dr 

By  the  principle  of  the  conservation  of  energy,  dlVl  =  dWz. 

Hence 

VdSdV=  -dpdr 
or 

drjdV=  -  VdS/dp  (6) 

For  ordinary  charges  dS  /  dp  will  be  sensibly  independent  of 
V.  Hence  we  have  for  the  total  change  in  r  when  the  condenser 
is  charged  from  a  neutral  state  to  the  voltage  V, 


=  -dSldp 


f  VVdV  '=  \SV\-  i/S-dSldp)  (7) 

Jo 


Homogeneous  Isotropic  Dielectric.  If  the  dielectric  is  homo- 
geneous and  isotropic,  (7)  may  be  simplified.  For  in  this  case 
S  and  dS  are  proportional  by  the  same  factor  to  the  product 
of  the  permittivity  and  the  linear  dimensions  of  the  condenser. 


1/2 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


Hence  if  we  denote  by  L  the  length  of  any  line  drawn  in  the 
dielectric,  we  have 

-  ijS'dSldp=-  ijcL-d(cL)ldp 

=  -  1 1  c- del  dp  —  ilL-dLldp=(kp  +  bl$) 

where  b  =  the  coefficient  of  compressibility  of  the  dielectric 
=  —sJL-dLjdp,  and  kp=  —  I  j  c  •  del  dp  =  the  coefficient  of  di- 
minution with  pressure,  or  increase  with  traction,  of  the  permit- 
tivity c.  Thus  (7)  becomes 

Ar=JSF2(^  +  ^/3)  (8) 

In  order  that  (7)  or  (8)  may  hold   when  the  dielectric  is   a 
fluid,  the  dielectric  must  be  completely  surrounded  by  the  con- 


A  and  B  are 

the  conductors,  C 

the  dielectric,  of  the 

condenser. 


Fig.  65. 

denser  plates,  or  the  plates  must  be  so  arranged  that  they  are 
kept  apart  by  the  pressure  of  the  fluid  only,  and  sensibly  all  the 
tubes  must  be  contained  in  the  fluid,  as  in  Fig.  65. 

If  the  field  of  the  condenser  is  uniform,  J5F2  =  r±c£2,  and 
(8)  may  be  written,  on  division  by  r, 

AT/T-j^^  +  j/a)  (9) 

3.  Change  in  Length  of  a  Line  Normal  to  a  Uniform  Electric 
Field  in  a  Solid  Isotropic  Homogeneous  Dielectric  During  Electri- 
sation. Let  e  denote  the  thickness  of  the  dielectric  of  a  parallel 
plate  condenser  (the  plates  always  remaining  in  contact  with  the 
single  dielectric),  and  L  and  L'  the  lengths  of  the  edges  normal 
to  e  of  a  rectangular  prism  of  the  dielectric.  We  shall  find  the 
change  in  L  when  the  condenser  is  charged  to  the  voltage  V. 
The  dielectric  in  all  that  follows  will  be  supposed  homogeneous 
and  isotropic. 


ELECTROSTRICTION.  1  73 

(1)  The  voltage  V  being  kept  constant,  let  the  dielectric  be 
subjected  to  a  traction  in  the  direction  of  L,  the  stress  across  the 
area  L'e,  normal  to  L,  being  Qr     This  will  increase  the  energy 
of  the  prism  by  Q^dL,  and  will,  in  general,  alter  its  capacity  by 
an  amount  dS.     Now  let  the  length  L  -f  dL  remain  constant 
while  the  voltage  is  increased  by  dV.     This  will  increase  the 
energy  by  (S  +  dS)-  VdV.     The  total  increase  in  the  energy  is 

dWl  =  Q,dL  +(S+  dS)  VdV 

(2)  Let  the  condenser  be  brought  from  the  same  initial  state 
to  the  same  final  state  as  before  by  a  different  process.     First  let 
the  voltage  increase  by  dV,  while  L  remains  constant,  which  will 
increase  the  energy  by  SVdV  and,  in  general,  the  traction  by 
dQr     Then,  the  voltage  (V+  dV)  being  kept  constant,  let  L  be 
increased  by  dL,  which  will  increase  the  energy  by  (Ql  +  dQ^dL. 
The  total  increase  is 

dW2  =  SVdV+  (Ql  +  dQ^dL 
As  in  §  2,  dW2  =  dWv  hence 

dLjdV=  VdSjdQ^  (10) 

Since,  for  small  changes  at  least,  dS  /  dQl  must  be  sensibly 
independent  of  V,  (10)  gives  for  the  total  change  in  L  when  the 
condenser  is  charged  from  V  =•  o  to  V  '=  V, 


AZ  =  dS/dQl       VdV=  \  SV\\ 
Since  5  =  cLL'  /  e, 
\IS-dSjdQl=  il 


Moreover,    i  /Z/  •dLf/dQl=  i  je-dejdQ^     Hence   we   have, 
putting  dQ^  =  L'e-dqv  simplifying,  and  dividing  by  L, 

&L/L=icF2/e2-(i/c-dc/dgl+  ilL-dLjdq,}        (12) 

Now  1  1  L  •  dLjdq^  is  the  reciprocal  of  the  stretch  modulus,  and 
will  be  denoted  by  M.     Also,  ijc-  dc  /  dq^  is  the  coefficient  of  in- 


174         ELEMENTS  OF    ELECTROMAGNETIC    THEORY. 

crease  of  the  permittivity  with  traction  normal  to  the  lines  of 
displacement,  and  will  be  denoted  by  kr     Thus  (12)  becomes 


,)  (13) 

The  above  results,  deduced  for  the  uniform  field  of  a  parallel 
plate  condenser,  will  hold  good,  without  sensible  error,  for  a 
condenser  of  any  form,  such  as  a  cylindrical  or  spherical  con- 
denser, in  which  the  conductors  are  parallel  and  so  close  to- 
gether that  E  is  sensibly  of  the  same  magnitude  throughout  the 
dielectric. 

4.  Change  in  Length  of  a  Line  in  the  Direction  of  a  Uniform 
Electric  Field  in  a  Solid  Isotropic  Homogeneous  Dielectric. 

Making  use  of  the  parallel  plate  condenser  of  the  last  article, 
and  of  the  same  general  method,  but  applying  a  traction 
Q2  =  LL'q2  parallel  to  the  lines  of  intensity,  we  obtain 

dejdV=  VdSjdQ2  (14) 

(15) 


and 

=  -^cF2/e2-  \M(2r 


+i)-^2] 

where  r  denotes  Poisson's  ratio,  and  kz  =  I  /  c-dc  /  dq2  is  the  co- 
efficient of  increase  of  c  with  traction  parallel  to  the  lines  of  in- 
tensity. 

The  results  just  established  hold  good,  like  those  of  §  3,  for  a 
thin  condenser  with  parallel  plates  of  any  form. 

It  is  easy  to  see  that 

kp=2k,+k2  (17) 

5.  Theory  and  Experiments.  A  rigorous  treatment  of  the 
general  theory  of  electrostriction,  together  with  a  resume  of  most 
of  the  experimental  and  theoretical  investigations  upon  the  sub- 
ject, is  contained  in  a  recent  memoir  by  P.  Sacerdote  (Ann.  de 
Chim.  et  de  Phys.  (7),  20,  p.  289,  1900).  Satisfactory  experi- 


ELECTROSTRICTION.  1 7  5 

ments  upon  the  values  of  the  coefficients  kpt  kv  and  kv  as  well  as 
entirely  satisfactory  experiments  upon  the  quantities  AT/T,  keje 
and  AZ,/Z,  have  not  yet  been  performed  (See  the  Philosophical 
Magazine  and  Nuovo  Cimento,  1900—1902,  for  some  of  the  most 
recent  and  best  results). 

§  §  2-4  are  based  largely  upon  portions  of  the  above-mentioned 
memoir  by  Sacerdote,  with  simplifications. 


CHAPTER   VI. 
ELECTRIC   ABSORPTION.     ELECTRETS. 

1.  Electric  Absorption.  In  all  that  precedes  electric  displace- 
ment has  been  treated  as  a  perfectly  elastic  phenomenon  ;  that 
is,  the  relation  D  =  cE  (analogous  to  Hooke's  law)  has  been  as- 
sumed to  hold  universally  with  c  at  every  point  a  constant,  inde- 
pendent of  the  time.  On  this  assumption,  the  capacity  of  a 
condenser,  which  is  proportional  to  cy  would  be  invariable  with 
the  time  of  charging.  This  appears  from  experiment  to  be  ac- 
curately true  for  dielectrics  whose  homogeneity  is  perfect,  for 
example,  gases,  pure  paraffine,  and  pure  calc  spar ;  but  it  is  by 
no  means  true  in  general,  as  the  experiments  described  below 
demonstrate. 

Let  a  condenser  whose  dielectric  is  not  homogeneous,  with  its 
plates  connected  to  the  quadrants  of  an  electrometer,  be  charged 
to  a  given  potential  difference  and  then  insulated  from  the  bat- 
tery. The  potential  difference  will  gradually  diminish,  approach- 
ing a  limit  sometimes  considerably  below  its  initial  value.  If 
now  the  condenser  is  short-circuited,  the  potential  difference 
becomes  zero ;  but  it  gradually  reappears,  unchanged  in  sign  but 
much  smaller  in  magnitude,  after  the  condenser  is  again  insulated. 
The  remnant  of  the  original  charge,  whose  presence  is  proved 
by  the  existence  of  this  potential  difference,  is  called  a  residual* 
charge.  If  the  operation  of  short-circuiting  and  insulating  is 
repeated,  the  same  phenomena  recur,  the  potential  difference  de- 
veloped after  insulation  being  smaller  each  time  and  finally  be- 
coming insensible.  The  disappearance  of  the  phenomena  is  of 
course  hastened  by  the  "  leakage  "  of  the  condenser,  if  appre- 
ciable, arising  from  the  conductivity  of  its  dielectric,  from  the 

presence  of  moisture,  etc. 

176 


ELECTRIC  ABSORPTION.  ELECTRETS.       1 77 

If  the  condenser  is  charged  for  a  long  time  in  one  direction, 
then  for  a  much  shorter  time  with  the  poles  of  the  charging  bat- 
tery reversed,  and  then  short-circuited  and  insulated,  a  potential 
difference  similar  to  that  last  applied  will  first  appear,  reach  a 
maximum,  diminish  to  zero,  change  sign  and  continue  to  increase 
in  the  direction  of  the  potential  difference  first  applied. 

Or  the  following  equivalent  phenomena  may  be  observed.  On 
the  condenser's  being  connected  with  a  constant  battery,  its 
charge  usually  reaches  very  quickly  almost  its  final  value,  but 
the  charge  goes  on  gradually  increasing,  sometimes  considerably 
exceeding  its  initial  magnitude.  On  short-circuiting  the  con- 
denser most  of  the  charge  disappears  ;  but  after  insulation  for  a 
short  time  a  second  discharge  in  the  same  direction  may  be  ob- 
tained, and  so  on,  till  the  discharges  become  too  small  to  be 
perceptible. 

Also,  if  the  condenser  is  charged  for  a  long  time  in  one  direc- 
tion, then  for  a  much  shorter  time  in  the  opposite  direction,  and 
then  short-circuited  and  insulated,  a  residual  charge  (and  cor- 
responding discharges,  if  the  condenser  is  repeatedly  short-cir- 
cuited) similar  in  sign  to  the  last  charge  will  at  first  appear,  but 
will  be  succeeded  by  a  residual  charge  similar  in  sign  to  that  of 
the  charge  first  applied. 

The  appearance  of  the  residual  charge  in  all  the  above  described 
experiments  is  hastened  by  subjecting  the  condenser  to  mechan- 
ical shocks. 

Two  possible  explanations  of  the  phenomena  of  electric  absorp- 
tion, as  the  phenomena  just  described  are  called  on  account  of 
what  was  once  regarded  as  the  soaking  in  of  the  electric  charge 
with  the  time,  have  been  given. 

(i)  The  general  analogy  between  electric  strain  and  stress  and 
mechanical  strain  and  stress,  together  with  the  fact  that  absorp- 
tion does  not  occur  in  free  aether  or  in  gases,  whose  elasticity  is 
perfect,  and  is  very  marked  in  a  substance  like  glass,  whose  elas- 
ticity is  extremely  imperfect,  has  led  to  the  suggestion  that  elec- 
tric absorption  is  due  to  the  imperfect  electric  elasticity  of  the 


178          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

dielectrics  which  exhibit  it.  The  phenomena  of  electric  absorp- 
tion are  exactly  analogous  to  the  phenomena  of  elastic  after- 
action.  Thus,  if  the  spring  of  §  36,  I.,  is  not  perfectly  elas- 
tic (and  no  solid  body  is  perfectly  elastic)  the  elongation  (analo- 
gous to  electric  charge)  produced  by  a  certain  applied  force 
(analogous  to  e.m.f.  of  charging  battery),  equal  and  opposite  to 
the  elastic  return-force  of  the  spring  (potential  difference),  will 
not  remain  constant  with  the  time,  but  after  reaching  almost 
immediately  a  value  usually  very  near  the  final  value,  will  gradu- 
ally increase.  If  now  the  force  is  removed  (condenser  short- 
circuited),  the  elongation  will  not  become  zero  at  once,  but  the 
spring  can  exert  no  force  by  virtue  of  the  remaining  elongation 
(residual  charge  when  potential  difference  =  o).  If  now  the 
spring  is  clamped  (condenser  insulated),  the  elongation  gradually 
becomes  elastic  (residual  charge  becomes  available,  potential  dif- 
ference increases  from  zero),  and  the  spring  exerts  a  force  upon 
the  clamp  (residual  potential  difference).  If  the  clamp  is  re- 
moved (short-circuit),  the  elongation  will  again  suddenly  dimin- 
ish, and  so  on.  Also,  if  the  spring  is  clamped,  extended  for  a 
long  time,  and  then  compressed  for  a  much  shorter  time  (con- 
denser charged  successively  in  opposite  directions)  and  then  re- 
leased (short-circuit),  the  residual  compression  will  gradually  reach 
zero,  and  then  become  a  residual  elongation  which  will  diminish 
much  more  slowly  to  zero  (condenser's  dischargers  will  be  for  a 
short  time  in  one  direction,  then  for  a  much  longer  time,  until  the 
whole  residual  charge  has  disappeared,  in  the  opposite  direction). 
(2)  Maxwell  has  developed  a  theory  according  to  which  the 
phenomena  of  absorption  can  not  occur  if  the  dielectric  is  per- 
fectly homogeneous  throughout,  but  must  occur  whenever  the 
ratio  of  the  permittivity  to  the  conductivity  is  not  constant  for 
all  parts  of  the  dielectric,  even  if  none  of  the  constituents  alone 
exhibits  the  phenomena.  This  conclusion,  according  to  which 
electric  absorption  is  due  to  heterogeneity  of  structure,  is  sup- 
ported by  experiments  of  Rowland  and  Nichols,  Muraoka, 
and  others.  It  is  quite  possible  that  deviation  from  perfect  elas- 


ELECTRIC  ABSORPTION.  ELECTRETS. 


179 


ticity  is  closely  connected  with  heterogeneity  of  structure,  and 
that  the  two  explanations  are  not  independent  of  one  another. 
Thus  glass  is  extremely  heterogeneous,  possesses  very  imperfect 
elasticity,  and  shows  the  phenomena  of  absorption  in  a  marked 
manner. 

2.  Dielectric  Absorption  Hysteresis.  If  a  condenser  whose  di- 
electric is  absorbent  is  rapidly  charged  to  a  voltage  Vf  t  short- 
circuited,  charged  in  the  opposite  direction  to  a  voltage  —  F', 
short-circuited,  charged  again  to  voltage  V,  and  the  process  re- 
peated a  number  of  times  at  the  same  rate  (by  connecting  the 
condenser  to  the  poles  of  an  alternating  current  dynamo,  for  ex- 
ample), it  is  evident  from  what  precedes  and  the  principle  of 
symmetry  that  the  relation  between  the  charge  q  and  voltage  V  of 
the  condenser  may  be  represented  by  a  closed  symmetrical  curve, 
such  as  that  in  Fig.  66.  When  the  voltage  has  dropped  from  the 


Fig.  66. 

value  V  at  A  to  0  at  B,  a  residual  charge  OB  is  left,  and  dis- 
appears entirely  only  when  the  voltage  reaches  the  negative 
value  OC.  As  the  voltage  increases  negatively  to  —  V  at  Dy 
the  charge  increases  negatively,  and  falls  to  the  value  OE  when 
the  voltage  again  becomes  zero.  The  residual  charge  again  dis- 
appears when  V=  OF—  —  OC,  etc.,  the  charge  thus  always 
lagging  behind  the  voltage. 


180         ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

If  however  the  condenser  is  carried  very  slowly  through  the 
cycle  of  charging,  discharging,  etc.,  or  if  the  dielectric  is  one 
which  does  not  exhibit  electric  absorption,  the  curve  is  found  to 
reduce  to  a  straight  line,  and  the  area  of  the  cycle  therefore  to  zero. 
(Cf.  Beaulard,  Journal  de  Physique  (3),  9,  422,  1900.) 

The  phenomena  are  therefore  not  analogous  to  those  of  mag- 
netic hysteresis  (§  39,  XIII.),  which  are  almost  wholly  indepen- 
dent of  the  time  in  which  a  cycle  is  completed  and  are  not  de- 
pendent, except  to  a  very  slight  extent,  upon  anything  similar  to 
viscosity  or  absorption.  The  term  hysteresis  may  be  used  to 
designate  the  electrical  phenomenon  described  in  this  article,  on 
account  of  the  lagging  effect  mentioned,  but  this  term,  if  so  used, 
should  be  coupled  with  the  word  viscosity  or  absorption  in  order 
to  avoid  the  incorrect  inference  that  the  phenomenon  is  physi- 
cally analogous  to  magnetic  hysteresis. 

3.  Energy  Dissipated  in  Dielectric  Hysteresis. — The  area  of 
the  curved  figure  ABCDEFA,  Fig.  66,  is 

H=fVdq  .,        „  (I) 

for  the  whole  cycle,  and  thus  represents  the  excess  of  the  elec- 
trical work  done  in  charging  the  condenser  (in  both  directions) 
over  the  electrical  energy  given  out  when  the  condenser  is  dis- 
charged (in  both  directions)  (see  §  37,  I.).  This  quantity  of 
energy  must  therefore  be  transformed  into  heat  during  each  com- 
pletion of  the  cycle, 
(i)  may  be  written 

H  =  J 'J *$EdL-dSdD  =  ffEdD-dr 

dL  and  dS  being  elements  of  a  line  of  intensity  and  an  equipoten- 
tial  surface,  respectively,  and  dr  being  the  element  of  volume 
dLdS.  We  have  therefore  for  the  energy  dissipated  per  unit 
volume  per  cycle  at  a  point  in  any  dielectric  where  the  intensity 
and  displacement  are  denoted  by  E  and  D, 

dHjdr  =  §EdD  (2) 


ELECTRIC  ABSORPTION.  ELECTRETS. 

the  integration    being  extended  throughout   a  complete  cycle. 
See  §  38,  I. 

4.  Intrinsic  Displacement  and  Intensity,  Electrets,  etc.  A 
dielectric  electrised  or  retaining  its  electrisation,  like  the  dielec- 
tric of  a  condenser  after  absorption  has  occurred,  partially  or 
wholly  under  the  action  of  internal  forces,  no  external  field,  and 
therefore  no  potential  difference  or  field  intensity  (E  =  —  dV  jdL) 
within  the  dielectric  itself  (as  when  the  condenser  is  short-cir- 
cuited), being  necessarily  present,  is  said  to  possess  intrinsic 
electrisation  or  displacement,  and  to  be  under  the  action  of  an  in- 
trinsic electric  intensity  or  force,  denoted  by  e,  in  the  direction  of 
the  displacement.  A  dielectric  in  this  state  is  called  an  electret. 

The  intensity  E=  —  dVj  dL,  §  3,  is  zero  at  two  points  of  the 
cycle  for  which  D  (the  total  displacement,  redefined  by  Gauss's 
theorem  *  as  dq[dS  at  a  conducting  surface,  the  theorem  being 
assumed  to  hold  for  intrinsic  as  well  as  for  elastic  displacement) 
has  finite  values,  while  D  (or  q)  is  zero  at  two  points  for  which 
E(QV  V}  has  finite  values. 

Thus  if  we  assume  the  relation  c  =  D  /  E  (by  which  D  was 
defined  in  the  case  of  elastic  displacement)  to  hold  in  the  case 
of  intrinsic  displacement,  r,  as  the  cycle  is  traversed,  will  pass 
through  all  values  from  +  oo  at  B  to  —  CD  at  E,  Fig.  66. 

If,  however,  we  introduce  the  conception  of  intrinsic  intensity 
e,  if  we  denote  the  field  intensity  —  dVj dL  by  E'  instead  of  E 
at  a  point  where  intrinsic  displacement  exists,  and  if  we  denote 
the  vector  sum  of  e  and  E'  by  (e  +  Ef )  =  E,  the  total  or  im- 
pressed intensity,  we  may  define  e  by  the  relation 

D  =  cE=c(e  +  E')  (4) 

With  this    understanding,    the    relation  D  =  c E  holds    univer- 

*  A  more  rigorous  and  general  definition  of  Z>,  analogous  to  the  general  definition 
of  £,  (63),  XIII.,  is  obtained  from  (2),  XV.  Thus 


(3) 

the  dielectric  being  in  a  neutral  state  at  the  time  t  =  o. 


1  82         ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

sally  and  leads  to  no  impossible  values  of  c,  since  e  and  D  have 
always  the  same  direction.  In  this  chapter  Ef  will  be  used  to 
denote  the  field  intensity  —  dVldL  ;  but  elsewhere,  except  where 
the  contrary  is  stated,  e  will  be  assumed  equal  to  o  and  Er 
equal  to  E. 

5.  Uniformly  Electrised  Spherical  Electret.  Suppose  the  elec- 
trisation of  the  sphere  of  §  12,  IV.,  to  become  partially  intrinsic. 
Then  let  the  external  charges  "producing"  the  (originally)  uni- 
form field  be  removed.  Then  the  only  remaining  "  charges  " 
are  the  fictitious  charges  upon  the  surface  of  the  sphere,  whose 

density  is  given  by  .        r        „ 

a'  =/cos  0  (5) 

where  J  denotes  the  intensity  of  the  remaining  electrisation. 

Inside  the  sphere  there  is  an  intrinsic  intensity  e  maintaining 
the  displacement  and  a  self-deelectrising  field  intensity  equal  to 

E'  =  -Jly,  (6) 

The  external  field  is  the  part  outside  the  sphere  of  the  field  con- 
nected with  the  doublet  of  moment 

Af-t-ir<tJ  (7) 

at  the   center  of  the   sphere. 

Maxwell's  plane  diagram  of  the  complete  field  is  given  in  Fig. 
67  (from  Maxwell's  Treatise,  §  143).  If  the  lines  of  displace- 
ment within  the  sphere  are  directed  from  5  to  Nt  the  lines  of 
intensity  have  the  direction  NS. 

The  quantities  M,  E',  o-f,  and  /can  all  be  expressed  in  terms 
of  the  internal  displacement  DQ  of  the  sphere. 

Thus  we  find  from  the  relations  (2)  IV.,  (94)  II.,  and  (7),  when 
6  =  o  and  R  =  a  :  DQ  =  MJ2  na3  ;  or 

(8) 
and 
(9) 


M=  2ira*DQ  =  I  TTtf3/  (8) 

The  relations  (i)  IV.,  (95)  II.,  and  (6)  give  when  6  =  90°  and 
=  a*  E>  =  MI^C^  =  -Dj2cl==  -7/3^  (9) 


ELECTRIC   ABSORPTION.     ELECTRETS. 


183 


The  relation/  =  D()  —  cfl  [(7),  IV.]  ,  or  either  of  the  last  two 
equations  alone,  gives 

J-Dt  +  Dtl2-\Dt  (10) 

From  this  equation  we  have 

<r'  =ycos#  =  f  £>0  cos  0  (ll) 


Fig.  67. 

The  magnitude  of  the  strength  of  each  pole  of  the  sphere  (dis- 
tributed over  a  hemisphere)  is 

q'  =  TTtf2/ =  1 7Trt2Z>0  =  f  n  (see  below)  (12) 

The  total  electric  flux  through  the  sphere  from  the  negative 
to  the  positive  pole,  and  back  again  outside  the  sphere  from  the 


1  84    ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

positive  pole  to  the  negative  pole,  all  the  tubes  of  displacement 
being  closed,  is 

f™y=f/  (13) 


The  total  flux  of  the  intensity  from  or  to  one  of  the  poles  is 


q'jc,  (  1  4) 


=  0      ,  -  = 

In  terms  of  the  intensities  and  the  permittivity  c2  of  the  sphere, 
the  internal  displacement,  denoted  above  by  DQJ  is 

D»  =  c2(c+E>)  (15) 

Since  e  and  E  have  opposite  directions,  DQ  is  less  than  if  e 
were  acting  alone.  By  short-circuiting  a  condenser  (making 
E  =  o)  after  undergoing  absorption  and  then  measuring  the 
residual  charge,  c2e  —  D'  =  intrinsic  displacement,  can  be  deter- 
mined, but  neither  quantity  can  be  determined  separately  except 
on  the  assumption  that  c2  is  the  same  for  intrinsic  as  for  elastic 
displacement. 

If  the  spherical  electret  were  placed  in  a  uniform  field  of  in- 
tensity £,  and  if  its  intensity  of  electrisation  /,  or  internal  dis- 
placement Z>0,  were  to  remain  rigidly  fixed  (cf.  §  8),  it  would  be 
acted  upon  by  the  same  forcive  as  that  which  would  act  upon  the 
doublet  of  moment  M  placed  at  its  center  in  the  same  field.  This 
forcive  is  easily  seen  to  be  a  torque 

T=  —  ME  sin  0  =  -  fccPJE  sin  6  =  -  27ia*DQ  E  sin  6    (16) 

in  the  direction  of  the  increase  of  0,  where  0  denotes  the  angle 
between  the  direction  of  electrisation,  or  the  axis  of  the  doublet, 
and  the  direction  of  the  uniform  field. 

The  same  result  could  of  course  be  obtained,  though  less 
simply,  by  integrating  the  expression  dT  =  Ea'dSxsm  0over  the 
surface  of  the  sphere,  where  x  denotes  the  distance  from  the 
equatorial  plane  to  the  element  of  area  dS  of  the  sphere. 

If  the  sphere  is  left  to  itself  after  the  removal  of  the  charges 
producing  the  uniform  field,  the  intrinsic  displacement,  and  there- 
fore the  internal  and  external  fields,  gradually  disappear.  This 


ELECTRIC    ABSORPTIONS.     ELECTRETS.  185 

gradual  disappearance  is  due  to  the  gradual  diminution  of  the 
intrinsic  intensity  and  the  continuous  action  of  the  self-deelec- 
trising  intensity,  which  acts  against  the  displacement,  or  tends  to 
reduce  the  apparent  surface  density.  Or,  better,  as  the  intrinsic 
forces  diminish,  the  tubes  of  displacement  are  gradually  freed 
and,  being  closed  tubes,  contract  to  nothing. 

If  the  sphere  is  covered  with  a  conducting  coat  while  the  in- 
ternal displacement  has  the  value  D^  the  external  field  disap- 
pears entirely  and  the  sphere  itself  is  left  in  the  same  condition 
as  the  dielectric  of  a  condenser  short-circuited  after  absorption 
has  taken  place.*  There  is  no  potential  difference  anywhere, 
but  the  intrinsic  displacement  within  the  sphere  has  increased, 
since  E'  ,  which  before  opposed  the  intrinsic  force  e  producing 
or  maintaining  the  displacement,  is  now  zero.  The  hemisphere 
which  before  had  a  positive  fictitious  charge  has  now  a  true 
negative  charge,  and  the  other  hemisphere,  before  apparently 
negative,  has  now  a  true  positive  charge.  The  law  of  the  dis- 
tribution of  the  true  charge  over  the  sphere  is  the  same  as  the 
law  of  the  distribution  of  the  previous  fictitious  charge.  If  the 
displacement  is  now  denoted  by  D'  =  c2e,  (15)  gives 

D>-cte-Dt-ctE'-[  (2Cl  +  c2)  1  2*,]  Dt 

=   [  (2^  + 


where  DQ,  J,  E'  are  the  values  of  the  displacement,  intensity  of 
electrisation,  and  self-deelectrising  intensity  immediately  before 
the  short-circuiting,  and  c.2  and  e  are  assumed  to  remain  constant 
during  the  process.  Since  the  final  value  of  the  self-deelectrising 
intensity  is  zero,  the  final  value  of  the  intensity  of  electrisation  is 
Jf  =  D'  .  D'  and  J'  are  wholly  intrinsic,  DQ  and  J  only  par- 
tially so.  The  density  of  the  true  charge  is 

er=  -  D'  cos  6  (18) 

No  change  is  produced  by  removing  the  conducting  cover. 
After  its  removal,  as  the  intrinsic  electrisation  continues  to  di- 

*See  Heaviside,  Electrical  Papers,  Vol.  I.,  p.  491. 


1 86          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

minish,  an  internal  and  external  field  of  the  same  character  as 
that  of  the  original  field,  but  opposite  in  direction,  is  developed. 
As  the  true  surface  density  everywhere  remains  constant  (in- 
sulation being  supposed  perfect),  while  the  intrinsic  electrisation 
gradually  diminishes,  this  field  will  grow  in  strength  ;  and  if  the 
intrinsic  displacement  could  disappear  entirely,  which  is  impossi- 
ble as  long  as  any  electric  field  remains,  the  sphere  being  ab- 
sorbent, the  total  field  would  finally  become  that  connected  with 
the  true  charges  given  by 

a  =  —  D'  cos  0 

only,  the  fictitious  charges  having  entirely  disappeared.  The 
displacement  and  intensity  would  now  be  in  the  same  direction 
at  any  point  within  the  sphere,  as  well  as  without. 

6.  Infinite  Circular  Cylindrical  Electret  Uniformly  Electrised 
Transversely.  In  exactly  the  same  way,  we  have  in  this  case  for 
the  apparent  surface  density 

<r'  =/cos  0=2D0cos6  (19) 

and  for  the  deelectrising  intensity,  or  intensity  due  to  the  ficti- 
tious charges,  within  the  cylinder 

E'  =  -Ji2C^-DJc,  (20) 

DQ  and  J  denoting  the  internal  displacement  and  intensity  of 
electrisation,  respectively. 

The  external  field  is  that  part  outside  the  cylinder  of  the  field 
of  the  line  doublet  of  moment 

M=  7ra?J  =  27ra2DQ  (2 1 ) 

placed  along  its  axis. 

The  plane  diagram  of  the  field  can  be  obtained  from  Fig.  28 
by  simply  drawing  a  circle  of  radius  a  with  the  center  of  the  dia- 
gram as  center,  annulling  all  the  lines  within  this  circle,  and  con- 
necting by  straight  lines  the  ends  of  each  circular  arc. 

The  magnitude  of  the  fictitious  charges  on  the  positive  and 
negative  halves  of  a  unit  length  of  the  cylinder  is 


ELECTRIC    ABSORPTIONS,     ELECTRETS.  l8/ 

q'  =  2aJ=  4aD0  =  2tt  (22) 

the  flux  across  unit  length  of  the  cylinder  is 

H  =  2aD0  =  fr'  (23) 

and  the  flux  of  intensity  from  or  to  the  fictitious  charge  upon 
unit  length  (either  half)  is 

IT  =  U/c,  -  2aE'  =  2U/C,  =  q'jc,  (24) 

7.  Natural  Electrets.    Pyroelectric  Crystals.    Kelvin's  Theory.* 

A  state  of  intrinsic  electrisation  exists  naturally  in  certain  crystals, 
for  example,  tourmaline,  which  are  called,  from  the  thermal 
relations  described  below,  pyroelectric  crystals.  In  its  ordinary 
condition,  however,  after  remaining  some  time  at  a  constant  tem- 
perature, the  external  field  of  such  an  electret  has  disappeared, 
on  account  of  poor  insulation,  like  that  of  the  spherical  electret 
of  §  5  after  being  covered  with  a  conducting  coat.  The  electret 
has  now  a  positive  charge  at  one  end  and  a  negative  charge  at 
the  other,  terminating  the  tubes  of  intrinsic  displacement  (there 
is  no  elastic  displacement).  Altering  the  temperature  of  the 
electret  alters  its  intrinsic  intensity  and  state  of  electrisation,  and 
therefore,  if  the  surface  remains  sufficiently  well  insulated  to  re- 
tain its  charges  when  some  of  the  tubes  of  intrinsic  displacement 
become  free,  develops  an  external  field.  The  direction  of  this 
field  depends  on  the  direction  in  which  the  intrinsic  forces  and 
electrisation  alter  with  the  increase  or  decrease  of  temperature. 
In  tourmaline,  as  would  be  expected  in  every  case,  the  electri- 
sation decreases  with  temperature  increase.  Hence  by  heating 
tourmaline  a  field  directed  like  that  of  the  sphere  of  §  5,  after  be- 
ing short-circuited  and  then  left  insulated  for  a  time,  is  developed. 
If  the  insulation  is  not  perfect,  this  external  field  will  gradually 
disappear.  If  the  electret  is  now  cooled,  its  electrisation  will  in- 
crease and  an  external  field  opposite  to  the  former  field  will  ap- 
pear, and  then  gradually  disappear  by  conduction  when  the  tem- 

*See  Heaviside,  Electrical  Papers,  Vol.  L,  p.  493. 


1 88          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

perature  is  kept  constant.  The  intrinsic  displacement  of  a 
pyroelectric  crystal,  and  therefore  the  true  surface  charges,  can- 
not be  made  to  disappear  like  those  of  an  electret  whose  electri- 
sation is  due  to  absorption.  By  breaking  a  pyroelectric  sub- 
stance across  its  axis  of  electrisation,  however,  positive  and 
negative  apparent  charges,  and  an  external  field  connecting  them, 
without  true  charges,  may  be  developed. 

Piezoelectric  Crystals.  A  state  of  electrisation  accompanied  by 
internal  and  external  fields  similar  to  those  of  the  sphere  of  §  3 
is  produced  in  some  crystals,  called  piezoelectric  crystals,  by  com- 
pressing or  stretching  them,  and  disappears  when  the  compres- 
sion or  stretch  is  removed.  The  external  field  corresponding 
to  a  given  state  of  strain  may  disappear  by  surface  conduction, 
leaving  the  surface  with  true  charges,  like  the  electrets  described 
above.  If  the  state  of  strain  is  altered  after  this  condition  has 
been  reached,  an  external  field  is  developed  whose  direction  de- 
pends on  the  direction  of  alteration  of  the  strain.  Some  crystals, 
like  tourmaline,  are  both  pyroelectric  and  piezoelectric. 

8.  Permanent  Electret.  The  fictitious  charge,  or  pole  strength, 
of  half  of  a  symmetrical  isolated  electret  is  not,  as  we  have  seen 
in  two  particular  cases,  §  §  5  and  6,  equal  to  the  flux  through  the 
electret,  but  is  greater  than  this  flux.  Thus,  although  the  fic- 
titious charge  upon  half  of  an  originally  neutral  sphere  placed  in 
a  uniform  field  is  less  than  the  flux  through  the  sphere,  the  fic- 
titious charge  upon  half  of  an  isolated  spherical  electret  uniformly 
electrised  is  one  and  one  half  times  as  great  as  the  flux  through 
the  sphere  ;  and  the  fictitious  charge  upon  half  the  surface  of  an 
isolated  infinite  cylindrical  electret  uniformly  electrised  trans- 
versely is  twice  as  great  as  the  flux  through  the  cylinder.  The 
infinite  cylinder  may  be  regarded  as  an  ellipsoid  of  revolution 
about  an  infinite  axis  perpendicular  to  the  direction  of  the  electri- 
sation, and  the  sphere  may  be  regarded  as  an  ellipsoid  with  its 
three  axes  equal.  We  shall  see  below  that  in  the  case  of  a  cir- 
cular cylinder  whose  length  is  very  great  in  comparison  with  its 


ELECTRIC    ABSORPTION.     ELECTRETS.  189 

diameter  and  which  is  electrised  longitudinally,  the  pole  strength 
at  either  end  is  approximately  equal  to  the  flux  through  the 
cylinder.  As  the  ratio  of  the  length  of  the  cylinder  to  its  diam- 
eter approaches  infinity,  the  ratio  of  the  pole  strength  of  either 
end  to  the  electric  flux  through  the  cylinder  approaches  unity 
indefinitely.  In  the  limit  we  have  an  ellipsoid  of  revolution 
about  an  infinite  axis  parallel  to  the  internal  displacement.  Thus 
the  greater  the  ratio  of  the  axis  parallel  to  the  electrisation  to 
the  other  axes,  the  more  nearly  does  the  pole  strength  equal  the 
electric  flux  across  a  pole  or  through  the  electret 

It  is  clear  that  if  an  electret  is  brought  into  the  field  of  an 
electric  charge  or  another  electret,  the  distribution  as  well  as  the 
strength  of  each  of  the  electret' s  poles  (or  each  of  the  poles  of 
both  electrets)  will,  in  general,  be  altered. 

Moreover,  if  the  medium  surrounding  an  electret  is  replaced, 
in  whole  or  in  part,  by  a  medium  of  different  permittivity,  the 
flux  through  the  electret,  and  therefore  the  fictitious  charges  or 
pole-strengths,  will  increase  or  decrease,  as  well  as  change  in 
distribution,  according  as  the  permittivity  of  the  new  medium  is 
greater  or  less  than  that  of  the  old  medium.  For  the  displace- 
ment, internal  and  external,  is  maintained  by  the  intrinsic  forces 
within  the  electret,  which  remain  constant  or  appreciably  con- 
stant during  the  change,  independently  of  the  surrounding  me- 
dium, and  must  produce  a  greater  or  less  flux  the  greater  or 
less  the  permittivity.  An  increase  of  the  same  kind,  and  greater 
in  extent,  occurs  when  the  external  field  is  destroyed,  the  sur- 
rounding medium  being  made  conducting  (or  the  permittivity 
infinite). 

The  energy  of  the  electret' s  field  may  be  divided  into  two 
parts,  the  energy  within  the  electret,  mostly  energy  of  intrinsic 
displacement,  and  the  energy  of  the  external  medium.  It  is 
clear  that  the  greater  the  ratio  of  the  intrinsic  energy  of  the 
electret  to  the  energy  of  its  external  field,  the  less  will  the  elec- 
trisation of  the  electret  be  affected,  either  in  distribution  or  in 
amount,  by  changes  in  this  external  field  (resulting  from  changes 


190    ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

in  the  medium) ;  also  the  greater  the  ratio  of  the  energy  density 
of  the  intrinsic  electrisation  to  the  energy  density  of  the  external 
field,  the  less  will  the  electret  be  affected  by  the  introduction  of 
other  electrets  or  charges. 

Consider  now  the  ideal  case  of  a  cylindrical  electret  of  very 
small  cross-section  and  great  length  uniformly  electrised  in  the 
direction  of  its  length,  except  within  very  small  regions  close  to 
its  ends  in  which  the  flux  diverges  or  converges.  The  poles  are 
approximately  concentrated  at  the  ends  of  the  electret,  and,  ex- 
cept within  the  very  small  volume  occupied  by  the  electret  itself, 
the  electric  field  is  similar  to  the  field  surrounding  two  electric 
charges  approximately  concentrated  at  a  distance  apart  equal 
to  the  length  of  the  electret.  If  the  length  of  the  electret  is 
very  great,  the  external  field  around  each  pole  is  practically 
radial. 

Since  the  energy  density  at  any  point  of  the  surrounding 
medium  is  proportional  to  the  square  of  the  intensity,  and  since 
the  intensity  is  inversely  proportional  to  the  square  of  the  dis- 
tance from  a  pole  (provided  the  distance  is  small  in  comparison 
with  the  length  of  the  electret),  the  energy  of  the  external  me- 
dium is  confined  almost  wholly  to  small  regions  surrounding  the 
two  poles.  If  the  length  of  the  electret  is  increased  while  the 
intensity  of  its  electrisation  is  kept  constant,  the  external  energy 
will  therefore  remain  very  nearly  constant. 

The  energy  of  the  uniform  intrinsic  electrisation,  however,  is 
proportional  to  the  length  of  the  "electret  for  a  given  value  of  the 
internal  electrisation  or  displacement. 

Hence  by  increasing  the  length  of  the  electret,  and  keeping 
the  flux  and  therefore  the  pole  strengths  constant,  the  ratio  of 
the  energy  of  the  intrinsic  electrisation  to  that  of  the  external 
medium  may  be  greatly  increased. 

Hence  such  an  (ideal)  long  slender  longitudinally  electrised 
electret,  if  made  of  a  substance  with  intrinsic  energy  density  very 
great  for  a  given  intensity  of  electrisation  (which  would  be  called 
an  electrically  hard  substance),  would  be  approximately  a  per- 


ELECTRIC  ABSORPTION.  ELECTRETS.        19! 

inanent  electret,  its  internal  energy,  electric  flux,  poles,  and  pole 
strengths,  being  practically  independent  of  the  external  field  (un- 
less the  external  field  should  be  destroyed,  when,  although  the 
internal  energy  and  flux  would  remain  sensibly  constant,  the  pole 
strengths  would  be  reduced  to  zero). 

The  force  between. such  a  pole  (concentrated)  and  an  extremely 
small  body  with  (concentrated)  charge  q  distant  L  therefrom 
would  be 

F=  qq'14-jrcL2  (25) 

where  qf  is  the  pole  strength  and  is  constant  (at  a  given  tempera- 
ture), and  c  is  the  permittivity  of  the  surrounding  medium. 

Since  the  volume  of  the  electret  is  negligible,  and  the  flux  from 
each  pole  in  the  external  medium  radial,  the  reaction  between  the 
two  fields  and  therefore  the  force  between  the  pole  and  the 
charged  body  must  be  the  same  as  the  force  between  a  very 
small  body  with  concentrated  true  charge  equal  to  IT,  the  flux 
through  the  electret,  and  the  small  body  with  concentrated 
charge  q.  That  is, 

F=gU/47rcL2  (26) 

On  comparing  (26)  with  (25),  we  see  that 

9'  =  n  (27) 

or,  the  flux  through  an  (ideal)  extremely  slender  longitudinally 
electrised  electret  of  great  length  is  equal  (strictly,  sensibly 
equal)  to  the  fictitious  charge,  or  pole  strength,  at  either  end. 

It  is  clear  from  what  precedes  that  any  of  the  electric  fields 
described  in  preceding  chapters  would  remain  sensibly  unaltered 
if  each  concentrated  true  charge  were  replaced  by  the  concen- 
trated pole  of  an  (ideal)  permanent  electret  of  very  great  length 
and  negligible  cross-section,  and  with  pole  strength  or  longitudi- 
nal flux  equal  to  the  charge  replaced. 


CHAPTER   VII. 

SPECIFIC    INDUCTIVE   CAPACITY.     THE    COMPARISON   OF 
PERMITTIVITIES. 

1.  Specific  Inductive  Capacity.  The  specific  inductive  capacity 
of  a  substance  is  defined  as  the  ratio  of  its  permittivity  to  the 
permittivity  of  the  standard  medium.  If,  as  in  this  book,  free 
aether  is  chosen  as  the  standard  medium,  the  specific  inductive 
capacity  of  a  dielectric  is  numerically  equal  to  its  permittivity 
(measured  in  the  electrostatic  systems  of  units,  XIV.),  since  <r? 
=  i  (in  the  electrostatic  systems). 

The  Comparison  of  Permittivities,  or  the  Determination  of  Spe- 
cific Inductive  Capacity.  Four  general  methods  of  comparing 
permittivities  will  be  considered  here  : 

I.  The  permittance  of  a  dielectric  bounded  by  a  fixed  system 
of  conductors  is  proportional  to  its  permittivity.      Hence  if  the 
whole  field  is  filled  in  succession  with  two  dielectrics,  and  the 
two  capacities  compared  experimentally,  the  ratio  of  the  permit- 
tivities will  be  known.      If  the  condenser  contains  two  different 
dielectrics  at  the  same  time  in  one  of  the  experiments,  the  method 
may  still  be  used  in  certain  simple  cases,  with  little  modification. 
See  §§  /and  8,  IV. 

II.  The  forcive  between  two  given  conductors  is  proportional 
to  the  permittivity  of  the  dielectric  filling  the  field  if  the  voltage 
is  kept  constant,  and  inversely  proportional  to  the  permittivity 
if  the  charges  are  kept  constant.      Hence  by  keeping  the  voltage 
constant  and  comparing  the  forcives  when  the  field  is  filled  with 
two  dielectrics  in  succession  the  ratio  of  the  permittivities  may 
be  determined.     The  comparison  by  means  of  constant  charges 
is  in  general  impracticable.     When  the  field  contains  two  dielec- 

192 


SPECIFIC   INDUCTIVE   CAPACITY.  193 

tries  at  the  same  time,  as  in  §  7,  IV.,  the  method  is  still  appli- 
cable, with  slight  modification,  in  certain  simple  cases.     See  §  2. 

III.  The  forcive  upon  a  dielectric  of  permittivity  c2  bounding, 
or  surrounded  by,   another  dielectric  of  permittivity  cv   in  an 
electric  field  depends  upon  the  ratio  of  c2  to  cr     Thus  by  meas- 
uring F  in   §  n,  IV.,  c2jcl  may  be  determined.     Two  methods 
based  upon  this  principle,  one  for  liquids,  and  the  other  for  solids 
(or  fluids  contained  in  a  vessel  made  of  a  solid  dielectric  of 
known  permittivity),  are  described  in  §§  3  and  4. 

IV.  When  lines  of  displacement  are  refracted  in  passing  from 
one  dielectric  to  another,  tan  0Jta.n  62  =  cjcr     Hence  by  measur- 
ing Ol  and  02,  the  ratio  cjc2  may  be  determined.     This  method  is 
discussed  in  §  5. 

From  §§  1-2,  VI.,  it  is  clear  that  the  .permittivity  ( if  defined 
as  DjE)  of  most  dielectrics  depends  to  a  greater  or  less  extent 
upon  the  time  of  electrisation,  being  greater  the  greater  the 
time,  up  to  a  certain  limit,  and  on  the  previous  history  of  the 
dielectric  (cf.  the  curve  in  Fig.  66),  except  for  slow  processes. 

From  §§  2-5,  V.,  it  follows  that  the  permittivity  depends  to 
some  extent  upon  the  stresses  in  the  dielectric,  which  may  be 
produced  wholly  by  electrical  causes,  unless  the  coefficients  there 
defined  vanish. 

2.  Method  II.  With  the  Quadrant  Electrometer.  Since  S,  §  5, 
III.,  is  proportional  to  c,  0  is  also  proportional  to  c  for  given 
values  of  VAy  VBy  and  VAE.  Hence  by  submerging  the  quadrants  in 
two  dielectrics  of  permittivities  cl  and  c2  successively  and  measur- 
ing the  resulting  deflections  for  the  same  voltages,  we  have 

V         'J'l-Wl  |     (0 

With  the  Kelvin  Absolute  Electrometer  or  the  Bichat  and  Blond- 
lot  Electrometer  In  the  same  way,  by  submerging  the  conduc- 
tors of  either  of  these  electrometers  in  two  dielectrics  in  succes- 
sion and  measuring  the  corresponding  values  of  Ft  §  4,  III.,  we 
have,  for  constant  voltages, 

cjc,  =  FJF,  (2) 


194         ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

In  the  case  of  the  Kelvin  electrometer  the  force  may  be  kept 
the  same  in  the  two  experiments  and  the  comparison  made  by 
altering  d.  In  this  case,  if  d^  and  d2  denote  the  values  of  d  when 
the  first  and  second  dielectrics  are  in  the  field 

,1;;JV:"::,/,    -:.:,         ',/',- W         I.Jv^S       (3) 

By  (25),  IV.  and  (3),  III.  the  Kelvin  electrometer  may  also  be 
used  when  one  of  the  dielectrics  is  in  the  form  of  a  plane  slab  of 
a  given  thickness  dz(<  d).  The  equations  corresponding  to  (4), 
III.  and  (5),  III.  are  easily  developed  for  this  case. 

When  any  one  of  these  instruments  is  used  idiostatically, 
alternating  as  well  as  unidirectional  voltages  may  be  applied, 
and  the  permittivity  thus  determined  when  the  time  of  electrisa- 
tion is  very  short. 

3.  Method  III.  duincke's  Method  for  Liquids.  Fig.  68  is  a 
diagram  of  the  apparatus.  A  and  B  are  the  conductors  of  a 
parallel  plate  condenser  separated  by  a  distance  d  very  small  in 
comparison  with  their  breadth  and  length,  and  immersed  in  a 
liquid  D  whose  permittivity,  cv  is  to  be  compared  with  that  of  a 


Fig.  68. 

gaseous  dielectric,  as  dry  air,  of  permittivity  cr  A  tube  E  com- 
municates through  a  small  opening  in  A  near  its  center  with  the 
region  between  the  two  plates.  This  tube  is  continuous  with  a 
manometer  tube  F  and  communicates  with  a  bulb  containing  dry 
air  by  the  stop-cock  G.  The  manometer  tube  contains  a  liquid 
of  density  p.  Its  cross-section  will  be  denoted  by  A. 

In  performing  an  experiment  a  wide  flat  air  bubble  C  is  first 
formed  between  the  plates  by  opening  G  and  pressing  the  air 
bulb.  G  is  then  closed,  and  the  difference  of  level  between  the 


SPECIFIC    INDUCTIVE   CAPACITY.  195 

two  liquid  surfaces  in  F  is  read.  If  this  difference  of  level  is 
denoted  by  hf  ',  and  if  the  acceleration  of  gravity  is  denoted  by  g, 
the  excess  of  the  pressure  in  C  (due  to  the  hydrostatic  pressure 
of  the  liquid  D  and  the  capillary  pressure  inward  at  the  edges  of 
the  air  bubble)  over  the  atmospheric  pressure  is  h'Apg.  The 
condenser  is  now  charged  to  a  voltage  V.  The  electric  intensity 
between  the  plates  is  uniform,  except  near  their  edges  and  near 
the  edges  of  the  bubble,  and  equal  to  E=  V  j  d.  The  electric 
pressure  (§41,  I.)  within  the  uniform  part  of  the  field  of  the  dielec- 
tric D  is  \c2Ez,  while  the  electric  pressure  within  the  uniform 
part  of  the  field  of  the  air  bubble  C  is  ^E2.  Hence,  if  cz  is 
greater  than  cv  the  bubble  will  contract  until  sufficient  air  has 
been  forced  out  into  the  manometer  tube  to  increase  the  differ- 
ence of  level  by  /z,  and  the  gaseous  pressure  by  hApg,  where 


when  there  will  again  be  equilibrium.      Hence  by  observing  h, 
A,  p,  g,  and  E  =  Vj  dyc2  —  cl  may  be  obtained  from  the  equation 

c2  -  c,  =  2kApglE*  =  2hApgd*l  V2  (4) 

In  what  precedes  we  have  assumed  the  capillary  pressure  of 
the  bubble  and  the  hydrostatic  pressure  of  the  liquid  D,  as  well 
as  the  total  volume  of  the  air,  to  remain  constant  throughout  the 
experiment.  We  have  also  assumed  the  slight  alterations  of 
gaseous  and  liquid  pressures  occurring  during  the  experiment  to 
bring  about  no  alterations  of  the  permittivities.  The  fact  that 
these  conditions  are  not  exactly  fulfilled  will  evidently  introduce 
no  sensible  error. 

(4)  Can  be  deduced  also  by  the  method  of  §  55,  I.,  from 
energy  considerations. 

4.  Method  III.  for  Solid  Dielectrics.  We  shall  consider  only 
the  simplest  case,  when  the  dielectric  is  in  the  form  of  a  plane 
slab.  Fig.  69  is  a  diagram  of  the  apparatus.  A  and  B  are  the 
two  conductors  of  a  parallel  plate  condenser  separated  by  a  dis- 


196          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

tance  d  small  in  comparison  with  their  length  and  breadth.  C  is 
a  plane  slab  of  the  dielectric,  of  permittivity  cv  hanging  in  air,  of 
permittivity  cv  with  its  sides  parallel  to  A  and  B,  from  the  arm  of 
a  balance  D.  The  thickness  of  the  slab  will  be  denoted  by  d2 
and  its  width  (perpendicular  to  the  plane  of  the  paper)  by  L. 
C  is  first  balanced  by  adding  weights  to 

A  /\ 

/A  the  scale  pan  of  D  while  A  and  B  are  at 
the  same  potential.  A  and  B  are  then 
charged  to  a  voltage  K  This  will  produce 
a  straight  field  between  A  and  B,  except 
near  their  edges  and  near  the  edges  of  the 
slab  C,  and  will  disturb  the  equilibrium. 
Equilibrium  is  then  restored  by  adding 
weights  to  the  scale  pan  if  c2  is  greater 
than  cv  or  by  removing  weights  there- 
*  from  if  c2  is  less  than  cv  as  follows  from 

§§7  and  10,  IV.,  or  from  §  7,  IV.,  and 
§55,  I.  Let  F  denote  the  downward  force 
upon  C  due  to  the  charging  of  the  con- 
denser. F  can  be  found  at  once  by  the 

method  used  in   §  3,  or   can   be  determined  as  follows  by  the 
method  of  §  55,  I. 

Imagine  C  to  suffer  an  infinitesimal  displacement  dx  downward 
from  its  equilibrium  position.  This  will  increase  the  cross-section 
of  the  uniform  part  of  the  field  through  C  and  air  by  Ldx,  and 
will  diminish  the  cross-section  of  the  uniform  part  of  the  field 
passing  through  air  only  by  the  same  quantity.  The  energy  of 
the  weak  field  outside  the  condenser  and  that  of  the  non-uniform 
field  near  the  edges  will  remain  sensibly  constant,  and  the  non- 
uniform  field  at  the  edges  of  C  will  move  unaltered  with  C. 
Hence  the  only  appreciable  change  in  the  energy  of  the  field  is 
that  due  to  the  fact  that  the  cross-section  of  the  uniform  field 
with  two  dielectrics  has  increased  by  Ldx,  while  that  of  the  uni- 
form field  through  air  only  has  decreased  by  the  same  quantity. 
Hence  the  total  increase  in  energy  is  sensibly 


SPECIFIC    INDUCTIVE   CAPACITY. 


197 


dW 


Hence 


F  = 


-  (c,  - 


(5) 


When  L,  cv  F,  V,  d,  and  dz  are  known,  cz  can  be  determined 
from  this  equation. 

In  obtaining  this  result  we  have  assumed  the  field  uniform 
throughout  the  whole  length  Z,  except  near  the  lower  and  upper 
edges  of  the  slab.  To  make  the  error  arising  from  the  non-fulfil- 
ment of  condition  negligible  L  must  be  great  in  comparison  with  d. 

5.  The  Method  of  Refraction  of  Lines  of  Displacement.  (Perot, 
Comptes  Rendus,  113,  p.  415,  1891.)  To  compare  two  permit- 


Fig.  71. 


tivities  by  this  method  it  is  necessary,  as  in  §§  3  and  4,  that  one 
of  the  dielectrics  be  a  fluid,  as  air.  The  principle  of  the  method 
may  be  developed  as  follows. 


195          ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

Let  a  large  triangular  prism  A,  Fig.  70,  of  permittivity  c2  and 
with  angle  a  be  placed  with  one  of  its  plane  sides  in  contact, 
or  otherwise  parallel,  with  a  large  metallic  plane  B\  and  let 
another  large  metallic  plane  C,  separated  from  A  (or  A  and  the 
other  conductor)  by  a  fluid  dielectric  D,  as  air,  of  permittivity  rp 
be  arranged  so  that  the  angle  0  between  C  and  the  nearer  face 
of  A  can  be  varied  and  measured.  If  B  and  C  are  charged  to  a 
voltage  Vy  the  electric  field  will  not,  in  general,  be  uniform  either 
in  A  or  in  D.  The  approximate  field  when  B  and  C  are  parallel 
is  shown  in  Fig.  71.  But  evidently  there  will  always  be  a  cer- 
tain value  of  /3  for  which  the  field  in  A  and  also  the  field  in  D 
will  be  uniform,  except  near  the  edges,  as  shown  in  Fig.  70.  In 
this  case  Ol  =  a  and  02  =  £  (§  2,  IV.).  Hence 

c2  jcl  =  tan  /3/  tan  a  (6) 

To  find  this  position,  a  very  thin  metal  plate  E  is  so  attached 
to  fine  insulating  threads  FF  as  to  be  movable  with  its  plane 
parallel  to  C  only.  If  the  field  in  D  is  uniform,  this  motion  will 
not  disturb  the  voltage  between  the  plates  ;  otherwise  the  voltage 
will  be  altered.  To  make  the  test,  then,  C  is  connected  to  earth 
(the  walls  of  the  room)  while  B  is  charged  to  potential  V  (poten- 
tial of  earth  =  o).  Then  the  condenser  is  insulated  and  C  is 
connected  with  the  electrode  of  an  electrometer,  the  other  pole 
of  which  is  to  earth.  Then  E  is  displaced  while  kept  parallel  to 
C.  If  the  electrometer  still  indicates  that  the  potential  of  C  is 
zero,  the  field  in  D,  as  well  as  that  in  A,  is  uniform.  If  not,  a 
second  adjustment  of  j3  must  be  made,  and  tests  and  adjustments 
repeated  until  the  potential  of  C  remains  zero,  or  until  the  dis- 
turbance of  its  potential  is  a  minimum,  when  E  is  displaced. 
Then  c2  jcv  can  be  determined  from  (6). 

The  close  agreement  between  the  values  of  <r2  jcl  found  by  this 
method  and  the  same  ratio  determined  by  other  methods  serves 
to  verify  the  correctness  of  (3),  IV.,  on  which  the  method  is  based. 


CHAPTER   VIII. 

THE   ELECTRIC    CURRENT.     THE   CONDUCTION   CURRENT. 

1.  The  Convection  Current.  If  a  small  light  conductor,  such 
as  a  gilded  pith  ball,  is  suspended  by  a  long  insulating  thread 
between  the  vertical  plates  A  and  B  of  a  charged  parallel  plate 
condenser,  it  will  fly  back  and  forth  between  the  plates  carrying 
opposite  charges  in  opposite  directions  and  gradually  discharging 
the  condenser. 

The  rate  dqjdt  at  which  positive  charge  is  carried  from  A  to  B, 
or  the  rate  dqjdt  at  which  negative  charge  is  carried  from  B  to  A, 
or  the  sum  of  the  two  rates,  dqjdt  +  dqjdt,  if  both  processes 
occur  simultaneously  (as  would  be  the  case  if  several  pith  balls 
were  present),  is  called  the  electric  convection  current  from  A  to  B, 
and  will  be  denoted  by  7cp.  That  is, 

/  p  =  dqjdt  +  dqjdt  =  dqldt  ( I ) 

Strictly,  the  convection  current  is  limited  to  the  space  actually 
occupied  by  the  moving  charge  or  charges. 

If  the  electric  volume  density  of  the  positive  charge  in  the 
element  of  volume  at  a  given  point  is  />,,  and  the  density  of  the 
negative  charge  in  the  element  p2  (the  element  containing,  in 
general,  both  positive  and  negative  charges),  and  if  the  velocity 
of  the  positive  charge  is  u^  and  that  of  the  negative  charge  in 
the  opposite  direction  «2,  then  the  convection  current  per  unit 
area  across  a  surface  normal  to  ul  in  the  element  of -volume,  or 
the  electric  convection  current  density  in  the  element,  is 

C  =  Pi«i  +  P2U2  (2) 

in  the  direction  of  ur 

199 


200  ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

2.  The  Dielectric  Current.  If  the  electric  flux  II  across  a 
surface  is  increasing  at  the  rate  dUjdt,  there  is  said  to  be  a  dielec- 
tric current  or  an  electric  displacement  current  through  the  surface 
equal  to 

(3) 


If  the  displacement  D  at  any  point  is  changing  at  the  rate 
y  the  dielectric  current  density  at  the  point  is 

(4) 

which  is  evidently  a  vector  with  the  direction  of  dD. 

According  to  our  mechanical  conception,  §  14,  I.,  the  dielec- 
tric current  in  free  aether  or  material  insulators  would  be  a  con- 
vection current  of  aether  cells  or  corpuscles. 

Convection  and  dielectric  currents  will  be  more  fully  discussed 
in  a  later  chapter  (XV.).  The  remainder  of  this  chapter  will  be 
devoted  principally  to  the  conduction  current. 

3.  The  Conduction  Current.  If  two  condenser  plates  A  and  B 
with  positive  and  negative  charges,  respectively,  are  connected 
by  a  wire  M,  the  electric  field  will  disappear  by  the  process  de- 
scribed in  §  42,  I.  During  this  process  there  is  an  electromo- 
tive force  along  and  through  the  wire  in  the  direction  AMB,  and 
the  wire  is  traversed  in  part  by  positive  charges  in  the  direction 
AMB  and  in  part  by  negative  charges  in  the  opposite  direction 
BMA.  The  wire  is  said  to  be  traversed  by  a  conduction  current. 
As  we  shall  see  later  (IX.,  §  15)  there  is  reason  to  believe  that 
the  conduction  current  consists,  in  the  general  case,  in  a  stream 
of  positively  electrified  particles  in  the  direction  of  the  e.m.f.  and 
a  stream  of  negatively  electrified  particles  in  the  opposite  direc- 
tion across  every  section  of  the  conductor. 

We  shall  define  the  strength  of  the  conduction  current,  or  the 
conduction  current,  /,  across  any  section  of  the  conductor  as  the 
rate  at  which  electric  charge  is  transferred  across  that  section  ; 
and  we  shall  define  the  direction  of  the  current  across  the  sec- 
tion as  the  direction  in  which  the  positive  charge  is  carried  across 


THE    CONDUCTION    CURRENT.  2OI 

the  section,  or  the  direction  opposite  to  that  in  which  the  nega- 
tive charge  is  carried.  If  both  positive  and  negative  charges  are 
simultaneously  crossing  a  given  section  in  opposite  directions, 
and  if  dq^  and  dq2  are  the  magnitudes  of  the  positive  and  nega- 
tive charges  carried  across  in  the  time  dt,  the  conduction  current 
across  the  section  is 

/  =  dqldt  =  dqjdt  +  dqjdt  (5) 

in  the  direction  of  transfer  of  the  positive  charge. 

When  the  charges  or  the  induction  in  (i),  (3),  or  (5)  are  ex- 
pressed in  RES  units  and  the  time  in  seconds,  the  electric  current 
is  said  to  be  expressed  in  the  RES  unit  current. 

A  method  of  measuring  the  electric  conduction  current  based 
on  the  above  definitions  is  described  in  §  3,  IX. 

In  the  case  of  an  ordinary  condenser  system  the  phenomenon 
of  discharge,  or  the  electric  current,  lasts  only  a  small  fraction 
of  a  second.  In  a  variety  of  ways  this  time  may  be  increased ; 
and  if  the  ends  of  the  wire  M,  instead  of  being  connected  to  the 
plates  of  a  condenser,  are  joined  to  the  terminals  of  a  voltaic 
cell,  or  other  agent  capable  of  maintaining  the  voltage  between 
its  ends  constant,  transient  effects  similar  to  those  described  in 
§  42,  I.,  will  at  first  occur,  but  a  steady  or  unchanging  state  will 
soon  set  in. 

4.  The  Conduction  Current  Density.  If  a  small  plane  area  dS 
is  imagined  within  the  substance  of  a  conductor  carrying  a  cur- 
rent, it  is  obvious  that  the  quantity  of  electric  charge  crossing 
dS  per  second  will  be  different  when  dS  is  turned  in  different 
directions,  and  will  be  a  maximum  when  the  normal  to  dS  points 
in  the  actual  direction  of  transfer  of  charge  at  the  point.  The 
ratio  of  the  current  dlc  crossing  the  area  dS,  with  its  normal 
turned  in  this  direction,  to  the  area  dS,  is  a  vector  called  the 
electric  conduction  current  density  at  the  point  considered,  and  will 
be  denoted  by  i .  Thus 

i.-dlJJS  (6) 

If  for  the  subscript  c  we  substitute  cv  or  d,  (6)  will  define  the 


202    ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

• 

convection  or  displacement  current  density  in  terms  of  the  con- 
vection or  displacement  current  consistently  with  §§  i  and  2. 

5.  Kirchhoffs  Law  I.  When  an  electric  conduction  current  is 
steady,  it  flows  in  a  closed  circuit  and  has  the  same  value  at 
every  section  of  the  conductor  which  carries  it.  For  if  the  cir- 
cuit were  not  closed,  or  if  the  current  across  any  two  sections 
were  not  the  same,  a  positive  or  negative  electric  charge  would 
continually  accumulate  at  either  end  or  in  the  region  between 
the  two  sections,  and  the  state  would  therefore  not  be  steady. 

This  proposition  will  be  extended  to  the  general  electric  cur- 
rent in  §  8,  Chapter  XV. 

An  obvious  (but  very  incomplete)  mechanical  analogue  of  the 
steady  conduction  current  is  the  flow  of  an  incompressible  liquid 
through  an  endless  pipe. 

KirchhofT's  law  L,  applied  to  the  unit  volume  of  a  conductor 
carrying  a  steady  current,  may  be  written 

div  ie  —  conv  ie  =  o  (7) 

since  the  current  entering  any  element  of  volume  across  one  part 
of  its  surface  is  equal  to  the  current  leaving  the  element  across 
the  rest  of  the  surface. 

Stream-tubes  and  Stream-lines.  From  (6)  and  (7)  it  is  now 
evident  that  a  steady  current  within  a  conductor  may  be  mapped 
out  by  a  system  of  lines  and  tubes  analogous  to  lines  and  tubes 
of  intensity,  etc.  These  tubes  and  lines  are  called  stream-tubes 
and  stream-lines,  or  lines  and  tubes  of  current  or  flow.  The  cur- 
rent density  at  any  point  has  the  direction  of  the  line  of  flow 
through  that  point ;  and  the  strength  of  the  current  across  every 
section  of  a  given  tube  is  the  same  and  equal  to  fic  dS  over  a 
diaphragm  5  normal  to  the  stream  lines. 

In  what  follows  we  shall  drop  the  subscript  c  and  denote  the 
conduction  current  and  current  density  by  /  and  i. 

6.  Electrodes.  Two  equipotential  surfaces  across  one  of  which 
the  current  enters  a  conductor  and  across  the  other  of  which  the 


THE   CONDUCTION   CURRENT.  203 

current  leaves  the  conductor  are  called  the  electrodes  of  the  con- 
ductor. The  electrode  by  which  the  current  enters  the  con- 
ductor is  called  the  anode,  and  that  by  which  the  current  leaves 
is  called  the  kathode. 

7.  Ohm's  Law  for  a  Steady  Current  in  a  Homogeneous  Iwfeepic 
Conductor  at  Uniform  Temperature,  Along  a  homogeneous  iee- 
trtirpH  conductor  at  uniform  temperature  throughout,  let  a  steady 
current  /  flow,  and  let  the  corresponding  voltage  between  two 
electrodes  in  or  terminating  the  conductor  be  denoted  by  F[2. 
Then,  as  a  result  of  experiment,  it  may  be  stated  that,  if  the 
voltage  or  current  is  varied  while  the  temperature  is  kept  con- 
stant (and  in  some  substances  at  least  certain  other  physical  con- 
ditions), the  current  /  is  proportional  to  the  voltage  PJ2.  That  is, 


The  proportionality  factor  K  is  called  the  conductance  of  the  por- 
tion of  the  conductor  between  the  given  electrodes,  and  its  recip- 
rocal, R  =  I  /  K,  the  resistance  of  the  conductor.  The  quanti- 
ties on  which  R  and  K  depend  will  be  discussed  below. 

Definitions  of  the  RES  unit  conductance  and  resistance  follow 
in  the  usual  manner  from  the  above  equations.  Thus  when  / 
and  V12  are  expressed  in  RES  units,  the  conductance  and  resis- 
tance also  are  said  to  be  expressed  in  RES  units. 

(8)  expresses  the  integral  form  of  Olnris  law  for  homogeneous 
is^tropic  conductors.  The  expression  of  the  law  for  non-homo- 
geneous circuits,  etc.,  will  be  developed  in  following  articles. 

8.  Conductance  and  Resistance  of  a  System  of  Conductors 
Connected  in  Multiple.  If  the  electrodes  of  any  number,  n,  of 
conductors  are  joined  together  so  that  the  anodes  form  a  com- 
mon anode  and  the  kathodes  a  common  kathode,  the  conductors 
are  said  to  be  connected  in  multiple.  Let  Kv  K2,  •  •  •  ,  Kn  and  R^ 
Rv  •  •  •  ,  Rn  denote  the  individual  conductances  and  resistances  of 
the  conductors,  Ilt  /2,  .  •  •  ,  7n  the  individual  currents,  and  /  the 
total  current  through  all  the  conductors,  when  the  voltage  be- 


204         ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

tween  the  electrodes  is^Jgj*  Z^.     Then  the  conductance  of  the 
system  is 


and  the  resistance  of  the  system  is 


=  i/(l/Rl+i/R2...+  i/Rn)  (10) 

9.  Conductance  and  Resistance  of  a  System  of  Conductors  Con- 
nected in  Series.  If  any  number,  n,  of  conductors  is  connected 
up  end  to  end  in  such  a  manner  that  each  surface  of  contact  be- 
tween two  conductors  is  an  equipotential  for  both  and  coincident 
with  the  original  electrodes  of  the  conductors  when  separate,  the 
conductors  are  said  to  be  connected  in  series.  Let  VQn  and  VQ1, 
F12,  •  •  •  ,  V{n_^n  denote  the  voltage  between  the  terminal  elec- 
trodes and  the  voltages  between  the  ends  of  the  successive  con- 
ductors, and  let  /  denote  the  current  along  all  the  conductors. 
Then  the  conductance  of  the  system  is, 


-I/OAK;  + 

and  its  resistance  is 

R=  1/^-^4-^  +•••  +  *„  (12) 

10.  The  Cylindrical  Homogeneous  Isotropic  Conductor.  Resist- 
ivity and  Conductivity.  Suppose  n  precisely  similar  right  cylin- 
drical conductors  joined  in  multiple  with  their  ends  as  common 
anode  and  kathode,  thus  making  a  cylindrical  conductor  of  n 
times  the  cross  -section  of  each  of  the  original  conductors.  If 
RQ  and  KQ  denote  the  resistance  and  conductance  of  each  cylin- 
der separately,  and  R  and  K  the  resistance  and  conductance  of 
the  system  of  n  conductors  in  multiple,  that  is,  of  a  cylinder  of 


THE    CONDUCTION    CURRENT.  205 

n  times  the  cross-section  with  its  ends  as  electrodes,  it  follows 
from  (9)  and  (10)  that 

K=  nKQ     and     R  =  RJn  (13) 

Thus  the  resistance  of  a  cylindrical  conductor  of  constant 
length  with  its  ends  as  electrodes  is  inversely  proportional  to  its 
cross-section,  or  its  conductance  is  directly  proportional  to  its 
cross-section. 

Suppose  the  n  conductors  connected  up  in  series,  thus  making 
a  cylindrical  conductor  of  the  original  cross-section,  but  of  n 
times  the  original  length.  If  R  denotes  the  resistance  of  the 
system  and  K  its  conductance,  ( 1 1 )  and  (12)  give 

K=KJn     and     R  =  nR0  (14) 

Thus  the  resistance  of  a  cylindrical  conductor  with  its  ends  as 
electrodes  is  proportional  to  its  length,  or  its  conductance  is 
inversely  proportional  to  its  length. 

Putting  the  two  above  results  together,  we  see  that  the  resist- 
ance R  of  a  cylindrical  conductor  with  its  ends  as  electrodes  is 
proportional  to  its  length  and  inversely  proportional  to  its  cross- 
section,  or  that  its  conductance  K  is  inversely  proportional  to  its 
length  and  directly  proportional  to  its  cross-section.  That  is,  if 
L  denotes  the  length  of  the  cylinder  and  A  its  cross-section, 

K=kA\L     or     R  =  rLjA  (15*) 

where  r  and  k  are  constants  depending  on  the  chemical  constitu- 
tion of  the  conductor  and  its  physical  condition,  r  is  called  the 
specific  resistance  or  resistivity  of  the  substance,  and  k,  its  recipro- 
cal, is  called  the  conductivity  of  the  substance.  (15)  may  be 

written 

r=RAjL     or     k=KLjA  (15*) 

When  R  and  K  are  expressed  in  RES  units  and  A  and  L  in 
c.g.s.  units,  r  and  k  are  said  to  be  expressed  in  the  RES  units 
resistivity  and  conductivity,  r  is  equal,  in  magnitude,  to  the  re- 


2C>6          ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

sistance  of  a  cube  of  the  substance  with  unit  edge  when  two 
opposite  faces  are  electrodes. 

11.  Differential  Form  of  Ohm's  Law.     From  the  preceding  arti- 
cle and  the  principle  of  symmetry  it  follows  that  in  the  case 
there  considered  the  electric  equipotential  surfaces  in  the  conduc- 
tor are  parallel  to  the  electrodes,  and  the  electric  intensity  E  uni- 
form and  parallel  to  the  length  of  the  conductor. 

From  the  same  article  it  also  follows  that  the  current  density 
is  uniform  throughout  the  conductor,  and  the  lines  of  flow  paral- 
lel to  the  length  of  the  cylinder,  and  therefore  coincident  with 
the  lines  of  intensity. 

Hence  we  may  substitute  in  (8)  for  F12  its  equal  ELy  for  7  its 
equal  Ai,  and  for  Kits  equal  kAj L\  then  Ai  —  kAjL-  EL,  or 

i=kE=Ejr  (16) 

In  the  case  considered  therefore  the  current  density  has  the 
same  direction  as  the  electric  intensity,  and  is  proportional  to  it 
in  magnitude.  Since,  moreover,  any  conductor  carrying  a  steady 
current  may  be  divided  up  into  elementary  tubes  of  intensity  and 
by  equipotential  surfaces  infinitesimally  distant  apart,  and  since 
the  cylindrical  volume  within  one  of  these  tubes  between  two 
successive  equipotentials  is  in  exactly  the  same  state  as  any  tube 
of  the  cylinder  considered  above,  (16)  is  seen  to  hold  in  general 
whether  the  tubes  are  straight  or  not. 

Since  at  the  surface  of  a  conductor  the  current  density  is  tan- 
gential to  the  surface,  the  electric  intensity  within  the  conductor 
at  the  surface  is  also  tangential. 

12.  The  Electric  Field  of  the  Steady  Conduction  Current.     Ex- 
cept as  stated  below,  the  electric  field  in  the  dielectric  surround- 
ing a  conducting  system  traversed  by  a  steady  electric  current 
has  all  the  properties  of  a  purely  static  field  connected  with  static 
charges  only.     The  tubes  of  displacement  terminate  at  the  sur- 
faces of  the  conductors  (if  homogeneous)  and  the  surface  charges 
at  their  ends  are  not  in  motion  and  take  no  part  in  the  conduction. 


THE  CONDUCTION  CURRENT.  2O/ 

For  within  the  conductor  (supposed  homogeneous)  div  i  —  div  kE 
—  k  div  E  =  o.  Hence  div  £  =  o  ;  and,  since  the  conductor  is 
homogeneous,  div  D  =  p  must  also  be  zero,  whatever  the  relation 
of  D  to  E  may  be.  Hence  within  the  conductor  the  positive  and 
negative  charges  per  unit  volume  at  any  point  are  equal.  Thus 
no  tubes  from  the  dielectric  penetrate  into  the  conductor,  but  all 
end  at  its  surface.  That  the  external  field  is  static  and  that  the 
surface  charges  take  no  part  in  the  conduction,  or  do  not  move, 
follow  from  the  consideration  that  the  field  surrounding  a  con- 
ducting system  traversed  by  a  steady  current  can  be  altered  in 
any  manner,  by  moving  the  circuit  or  by  bringing  up  charged 
bodies  insulated  from  it,  without  in  any  way  affecting  the  (steady 
value  of  the)  current.  Also  the  surfaces  of  insulated  conductors 
placed  in  the  field  are  equipotential  surfaces  and  traversed  by  no 
currents ;  hence  the  tubes  ending  upon  them  (and  connected 
with  the  current-carrying  conductors)  are  not  in  motion.  The 
same  thing  follows  from  Ohm's  law,  the  resistance  of  a  con- 
ductor not  being  a  function  of  its  external  surface,  as  it  would 
be  if  the  surface  charges  took  part  in  conduction. 

The  conductor  itself,  as  shown  in  §  1 1 ,  also  contains  an  electric 
field  invariable  with  the  time.  Little  or  nothing  is  known  of  the 
electric  displacement  in  good  conductors  traversed  by  steady 
currents. 

The  electric  field  within  and  without  the  conductor  is  accom- 
panied by  a  magnetic  field  (XI.  and  XII.)  and  is  the  seat  of  the 
transfer  of  energy  (XVI.),  while  the  conductor  is  also  the  seat 
of  the  dissipation  of  energy  in  heat  (§  15). 

The  lines  of  intensity  within  the  conductor  are  tangential  at 
the  surface,  as  shown  in  §  1 1 ,  and  the  lines  of  intensity  in  the 
dielectric  do  not  meet  the  conducting  surfaces  normally. 

Let  El  denote  the  electric  intensity  in  the  dielectric  just  out- 
side the  conductor  at  a  point  P  of  the  interface,  and  0l  the  angle 
made  by  El  with  the  normal  to  the  surface  of  the  conductor ;  and 
let  E  denote  the  intensity  just  within  the  conductor  at  the  same 
point  of  the  interface.  E,  as  already  shown,  is  parallel  to  the 


208          ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

interface.  By  the  principle  of  the  conservation  of  energy  we 
may  show,  just  as  in  §  2,  IV.,  that  El  and  E  are  in  the  same 
plane  normal  to  the  interface,  and  that 

£lSm0^£  (17) 

If  the  conductor  is  a  perfect  (imaginary)  conductor,  that  is,  if 
its  conductivity  is  infinite,  E  =  o,  since  otherwise  the  current 
would  be  infinite.  Hence  in  this  case  6l  =  o,  or  the  tubes  of 
displacement  in  the  dielectric  meet  the  surface  of  the  conductor 
normally. 

13.  The  Laws  of  Refraction  of  Stream-lines.  At  the  interface 
between  two  substances  of  different  conductivities  k^  and  /£2  a 
stream-line  of  a  steady  current  is  refracted  in  such  a  manner  that 
the  incident  and  refracted  lines  are  in  the  same  plane  perpen- 
dicular to  the  interface,  and  that 

tan(91/tan(92  =  ^1/^2  (18) 

where  6l  and  #2  denote  the  angles  made  with  the  normal  to  the 
interface  by  the  incident  and  refracted  portions  of  a  stream  line. 
For,  since  the  current  is  steady,  so   that  no  electric  charge 
accumulates  anywhere,  Kirchhoff's  law  I.  gives 

z'j  cos  6l  —  z'2  cos  02  (19) 
and  the  principle  of  conservation  of  energy  gives 

E^  sin  0l  =  E2  sin  02  (20) 

and  these  equations,  since  i  —  k  E,  give  ( 1 8)  on  division.  Cf. 

§  2,  IV. 

c  being  interchanged  for  k  and  i  for  D,  the  discussion  in  §  2, 
IV.,  and  the  description  in  following  articles  of  fields  in  two  or 
more  dielectrics  apply  to  the  fields  of  intensity  and  flow  in  con- 
ductors. It  must  always  be  remembered,  however,  that  one  of 
the  /£'s  may  be  zero,  while  c  can  never  be  less  than  CQ  =  i  ;  so 
that  kj k^  may  be  zero  or  infinity  without  either  kl  or  /£2's  being 
infinite. 


THE   CONDUCTION   CURRENT.  209 

14.  General  Formula  for  Conductance    and    Resistance,     Con- 
ductance and  Permittance.     From  (8)  we  have 


while  (6)  gives 

I-fulS 

the  integration  being  taken  over  an  electrode,  or  over  the  portion 
within  the  conductor  of  any  equipotential  surface.      Moreover, 


along  a  line  of  intensity  or  flow  from  one  electrode  to  the  other. 
Hence 

K=  i  JR  =  7/F12  =  fkEdSjfEdL  =  fkEdSjVl2     (21) 

By  comparing  (16)  and  (21)  with  (3)  and  (24),  Chapter  I.,  it 
will  be  seen  that  K  bears  the  same  relation  to  k  that  the  per- 
mittance 5  bears  to  the  permittivity  c.  Since  K=  fkEdS  JVIV 

while  vS  =  J  cEdS  \V^,  the  process  of  finding  the  conductance  of 
the  portion  of  a  conductor  between  two  given  electrodes  or  equi- 
potential surfaces  is  identical  with  that  of  finding  the  permittance 
of  a  dielectric  occupying  the  same  space  as  that  occupied  by  the 
conductor  and  having  the  same  electric  field  as  that  within  the 
conductor,  except  that  k  must  be  substituted  for  c.  In  most 
permittance  problems  it  is  impossible  to  deal  accurately  with 
finite  electrodes  and  finite  electric  fields,  there  being  no  substance 
of  zero  permittivity  with  which  an  electric  field  may  be  sur- 
rounded to  prevent  its  spreading  indefinitely.  A  conductor,  on 
the  other  hand,  may  easily  be  placed  in  a  region  of  zero  con- 
ductivity, so  that  the  current  tubes  are  wholly  restricted  to  its 
own  substance,  and  within  the  conductor  the  lines  of  flow  and 
lines  of  intensity  are  coincident.  This  makes  conductance 
problems  much  simpler  in  many  cases  than  the  corresponding 
problems  in  electrostatics. 

15.  The  Conductance  and  Resistance  of  Various  Conductors.  In 
all  the  examples  which  follow  the  electrodes  may  be  supposed 


210          ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

to  be  surfaces  of  infinitely  thin  sheets  of  perfectly  conducting 
material  (k  =  infinity),  in  order  to  insure  their  being  true  elec- 
trodes, that  is,  equipotentials  normal  to  the  lines  of  intensity  and 
flow  (see  §  12). 

The  corresponding  permittances  having  been  already  deter- 
mined (Chapter  II.),  the  conductances  are  found  as  indicated  in 
the  last  article,  from  the  following  relation 


(22) 

1.  For  a  right  cylinder  of  cross-section  A  and  length  d,  with 
its  ends  as  electrodes,  (29),  II.,  gives,  with  (22), 

K=ijR  =  kAld  (23) 

as  already  shown  directly  in  §  10. 

2.  For  a  conducting  spherical  shell  between  two  spherical  elec- 
trodes of  radii  L^  and  L2  =  L^  -f-  d,  (5),  II.  gives,  in  the  same  way, 

K=\IR-  (#*ki  d)  L?  (  i  +  rf/z,)  (24) 

For  a  hemispherical  shell,  half  of  the  last,  we  have 

K'  =  \K=^  (27rkjd)  L?(i+  <//£,)  (25) 

and  so  on  for  all  fractions  of  the  shell  obtained  by  cutting  it  up 
with  cones  having  their  apices  at  its  center. 

3.  For  the  conductance  of  an  infinite  conductor  in  which  two 
spherical  electrodes  of  radii  L^  and  L.2  are  immersed  at  a  great 
distance  apart,  a  slight  generalisation  of  (43),  II.  gives 

^=47r£/(i/Z1-f  i/Za)  (26) 

If  the  infinite  conductor  is  bounded  by  a  plane  surface,  and  if 
two  hemispherical  electrodes  are  placed  in  the  conductor  with 
their  bounding  circles  in  the  plane  of  the  conductor's  surface, 
the  conductance  is 

K'  =  lK=27rkj(ijLl+ilL,)  (27) 

4.  The  conductance  of  a  right  circular  cylindrical  shell   of 
radii  L  and  L  +  d  and  of  length  /  is,  from  (23),  II., 

K=  27r^//log  (i  +  djL)  [2S) 


THE    CONDUCTION    CURRENT.  211 

5.  The  conductance  of  an  infinite  plane  slab  of  thickness  /  and 
with  two  right  circular  cylindrical  electrodes  of  radius  R  and 
centers  distant  2d  apart,  is,  by  (62),  II., 

{R/  Id-  (<P  -  *«)»]  }  (29) 


If  the  slab  is  cut  symmetrically  into  halves  by  a  plane  passing 
at  right  angles  to  the  plane  through  the  axes  of  the  two  elec- 
trodes, and  if  this  plane  is  made  an  electrode,  the  conductance 
of  either  half  is  twice  that  of  the  whole  ;  or 

K'  =  2K=  27n£//log  {Rf  \d  -  (d2  -  R^~]  }  (30) 

In  the  same  manner,  from  Chapter  IV.,  the  conductances  of 
some  simple  non-homogeneous  conductors  may  be  obtained. 

16.  Joule's  Law  :  In  a  homogeneous  conductor  of  resistance 
R  traversed  by  a  current  /  heat  is  developed  at  the  rate 


This  relation  may  be  established  as  follows  :  In  the  time  dt  a 
charge  ^(+)  anc^  a  charge  dq^—}  equivalent  to  a  charge  Idt 
(-J-)  in  the  direction  of  the  current  cross  every  section  of  a  con- 
ductor carrying  a  steady  current  7.  Hence  the  work  done  in 
the  time  dt  by  the  electric  field  upon  the  conductor  of  resistance 
R,  if  the  voltage  between  its  electrodes  is  F12,  is 

dW  =  V^Jdt  =  RI-  Idt  =  RI*dt 

by  §  1  2  and  the  law  of  Ohm.  Hence  the  time  rate  at  which 
work  is  done  by  the  electric  field  in  the  conductor  of  resistance 
Ris 

dWldt=Vl2I=RI*  (32) 

Now  heat  is  always  developed  in  a  conductor  during  the 
passage  of  a  current.  Hence,  if  no  other  transformation  of 
energy  occurs,  between  the  electrodes  of  the  conductor  dW\dt  — 
RP  must  be  the  rate  at  which  heat  is  generated  in  the  con- 
ductor when  traversed  by  the  current  /.  That  this  is  the  case 


212          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

when  the  conductor  is  homogeneous  (including  constancy  of 
temperature  throughout)  has  been  proved  by  the  researches  of 
Joule  and  others,  whose  experimental  results  are  in  strict  accord 
with  the  above  equation  when  dWjdt  is  equated  to  dHjdt. 
Hence  the  law,  expressed  in  (31).  If  the  conductor  is  not 
homogeneous  (and-  for  all  but  three  directions  of  -the-  stream - 
1ir>o0|  »f  *^p  y»™»,jy/^?T  fa  «%mf  «v/yfcv^. )  ft  contains  intrinsic 

e.m.f.s,  §  19,  and  in  addition  to  the  Joulean  heat  transformation 
other  transformations  take  place. 

Differential  Form  of  Joule's  Law.  The  dissipativity  at  any 
point  of  a  conductor  is  the  time  rate  per  unit  current  (squared) 
at  which  heat  is  there  generated  per  unit  volume.  To  find  the 
dissipativity,  which  will  be  denoted  by  dhjdt,  consider  the  ele- 
mentary volume  enclosing  the  given  point  and  included  within  a 
tube  of  flow,  of  cross-section  dS,  between  two  equipotentials  dis- 
tant dL  apart.  The  resistance  of  this  element  of  volume  dr  = 
dL  dS,  is  EdLjidS  the  current  along  the  tube  being  idS. 
Hence 

dhldt  =  (EdLjidS]  (idS}* I (dSdL)  =  Ei  =  kE2  =  n'2     (33) 

which  expresses  Joule's  law  as  applied  to  the  element  of  volume 
at  any  point  of  a  conductor. 

Experiment  justifies  the  statement  that  (31)  and  (33)  are  ap- 
plicable to  any  conductor,  homogeneous  or  not,  the  total  heat  de- 
veloped being  equal  to  the  sum  of  the  Joulean  heat  and  the  heat 
developed  owing  to  the  operation  of  other  factors  than  resistance. 

By  the  last  equation,  (31)  may  be  written 

dHjdt  =  RP  =  fEidr  =  fkE2dr  =  JV2 jk'dr  =  frPdr    (34) 

the  integrals  being  extended  throughout  the  part  of  the  conductor 
considered.  From  this  equation  it  is  clear  that  the  amount  of 
heat  dissipated  per  unit  time  by  the  resistance  of  a  conductor  can 
be  obtained  from  the  formula  for  the  energy  in  the  correspond- 
ing case  in  electrostatics,  viz.,  W '=  §±cEzdrr,  by  substituting  k 
for  ±c,  etc. 


THE  CONDUCTION  CURRENT.  213 

17.  Definition  of  Resistance  by  Joule's  Law.  The  proportion- 
ality between  dHjdt  and  72  having  been  established  by  experi- 
ment, the  resistance  of  a  conductor,  R,  might  have  been  defined 

by  the  relation 

(35) 


without  recourse  to  Ohm's  law.  This  procedure  would  be  in 
perfect  harmony  with  all  that  is  known  of  the  nature  of  resist- 
ance, whose  only  function  seems  to  be  the  dissipation  of  energy 
in  heat,  as  it  is  dissipated  by  mechanical  friction. 

Joule's  Method  of  Determining  Resistance  in  Absolute  Measure. 
By  placing  a  conductor  in  a  calorimeter  and  measuring  the  rate 
dHjdt  at  which  heat  is  developed  therein  when  a  known  current 
/  traverses  the  conductor,  its  resistance  R  may  be  obtained  from 

(35). 

18.  Mechanical  Analogue  of  the  Law  of  Ohm  and  the  Law  of 
Joule.       Consider  a  pipe  through  which  an  incompressible  liquid 
flows  at  a  constant  rate,  the  volume  of  liquid  carried  per  unit  time 
across  every  section  of  the  pipe  being  /.     The  flow  of  the  liquid  is 
opposed  by  a'frictional  pressure  assumed  to  be  proportional  to 
/.      Let  this  pressure  be  denoted  by  —  RI,  R  being  a  constant 
for  the  given  pipe  and  liquid.     To  overcome  this  pressure,  that 
is,  to  keep  up  the  constant  rate  of  flow  7,  an  equal  and  opposite 
pressure  F12  =  +  RI  must  be  applied  in  the  direction  of  the  cur- 
rent.    This  pressure  does  work  against  friction  at  the  rate  V^I 
=  RI2,  which  is  therefore  the  rate  at  which  energy  is  dissipated 
in  heat  in  the  circuit. 

19.  Intrinsic  and  Impressed  Electromotive  Force.     In  order  to 
maintain  an   electric   current,  with   its   continual   dissipation   of 
energy  in  heat  according  to  Joule's  law,  and  its  possible  per- 
formance of  work  of  various  kinds,  every  circuit  continuously 
carrying  a  current  must  contain  one  or  more  regions  in  which 
energy  in  some  other  form,  as  mechanical,  chemical,  or  thermal 
energy,  is  transformed  into  the  energy  of  the  electric  current  (the 
energy  of  the  electromagnetic  field).     Such  a  region  is  said  to 


214          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

contain,  or  to  be  the  seat  of,  an  intrinsic  electromotive  force  ;  and 
the  agent  through  which  the  energy  transformation  is  effected, 
or  may  be  effected,  as  a  dynamo,  a  voltaic  cell,  or  a  thermo- 
couple, is  said  to  possess  the  intrinsic  electromotive  force. 

In  strictness,  the  intrinsic  electromotive  force  in  a  region  is 
defined  as  the  rate  at  which  energy  in  some  other  form  is  there 
transformed  into  the  energy  of  the  electric  current  (the  energy 
of  the  electromagnetic  field)  divided  by  the  strength  of  the  cur- 
rent. Thus,  if  P  denotes  the  rate  at  which  electrical  energy  is 
generated,  or  power  taken  into  the  circuit  in  the  region  by  trans- 
formation, I  the  current,  and  ^  the  intrinsic  electromotive  force 

•9  "PI  I  (36) 

Any  part  of  a  circuit  in  which  electrical  energy  is  transformed 
into  energy  of  another  form  is  also  said  to  contain  an  intrinsic 
e.m.f.,  provided  that  the  agent  effecting  the  transformation  when 
acting  independently  can  reverse  the  direction  of  the  transforma- 
tion, or  transform  energy  of  the  other  form  into  electrical  energy, 
i.  e.,  itself  maintain  an  electric  current.  Thus  an  electric  motor, 
by  which  electrical  energy  is  transformed  into  mechanical  energy, 
and  a  storage  battery  while  charging,  by  which  electrical  energy 
is  transformed  into  chemical  energy,  possess  intrinsic  e.m.f.s, 
since  each  acting  alone  can  generate  electrical  energy,  the  one 
when  mechanically  driven  acting  as  a  dynamo,  the  other  as  a 
voltaic  cell.  The  intrinsic  e.m.f.  T"  is  given  in  all  cases  by  (36), 
proper  attention  being  paid  to  the  sign  of  P.  Thus  if  in  any 
region  power  is  taken  into  the  circuit  by  transformation,  P  in  this 
region  is  positive  and  W  and  /  have  the  same  direction.  If  in  any 
region  electrical  energy  is  transformed  into  some  other  form  of 
energy,  or  power  given  out  by  the  electrical  system,  P  in  this 
region  is  negative,  ^  and  /  have  opposite  signs,  or  directions,  and 
the  intrinsic  e.m.f.  opposes  the  current.  It  is  by  overcoming 
this  counter  e.m.f.  that  the  transformation  of  electrical  energy 
into  energy  of  some  other  form  is  effected.  In  all  cases  a  reversal 
of  the  current  reverses  the  sign  of  the  energy  transformation  by 
an  agent  with  an  intrinsic  e.m.f. 


THE    CONDUCTION    CURRENT.  215 

Also,  any  region  in  which  energy  is  transformed  from  some 
other  form  into  electrical  energy,  or  from  electrical  energy  into 
energy  of  some  other  form,  at  the  time  rate  P  when  the  current 
has  the  value  7  is  said  to  contain  an  e.m.f.  Pj  /,  although  in  the 
latter  case  the  e.m.f.  may  not  be  intrinsic.  Thus,  according  to 
Joule's  law,  a  homogeneous  conductor  of  resistance  R  when  tra- 
versed by  a  current  /  transforms  electrical  energy  into  heat  at 
the  rate  RI2.  Hence  the  conductor  is  the  seat  of  an  electro- 
motive force  equal  to 

-  (dH\dt)\I-  -  */«//=  -  R  /=  -  F12  (37) 

This  negative,  or  counter,  e.m.f,  —  R 1=  —  F12  is  not,  however, 
included  among  intrinsic  e.m.f.s,  since  if  the  conductor  is  heated 
a  current  is  not  produced,  or  if  the  direction  of  the  current  is 
reversed,  heat  is  still  generated  at  the  same  rate,  not  absorbed,  as 
it  would  be  if  the  e.m.f.  were  reversible  and  intrinsic. 

Other  electromotive  forces  exist,  like  potential  differences 
(non-intrinsic  e.m.f.s  of  static  fields  or  fields  of  conductors  carry- 
ing steady  currents)  and  the  non-intrinsic  e.m.f.s  of  induction 
(XIII.),  by  which  energy  is  transferred  in  the  electromagnetic 
field,  but  never  transformed.  These  e.m.f.s,  together  with  in- 
trinsic e.m.f.s,  may  act  as  impressed  e.m.f.s.  The  impressed  e.m.f. 
between  the  electrodes  of  a  conductor  is  equal,  by  definition,  to 
the  sum  (§  21)  of  the  intrinsic  e.m.f.s  included  between  them, 
P 1 1,  plus  the  time  rate  Pf,  at  which  electromagnetic  energy  is 
transferred  to  the  region  between  them  from  the  surrounding 
field  (developed  by  intrinsic  e.m.f.s  in  other  parts  of  the  circuit  or 
in  other  circuits  transforming  energy  of  another  kind  into  electro- 
magnetic energy  if  P1  jl  is  positive)  divided  by  the  current  /. 

Thus  the  impressed  e.m.f.  in  a  homogeneous  conductor  of  re- 
sistance R  carrying  a  steady  current  /  is  FJ2  =  RI  (exactly 
equal  and  opposite  to  the  counter  e.m.f.  of  resistance,  —  /?/), 
the  difference  of  potential  F12,  by  which  the  energy  is  transferred 
to  the  conductor  at  the  rate  F12/=  RPy  being  developed  by  an 
intrinsic  e.m.f.  situated  outside  the  portion  of  the  circuit  consti- 
tuting the  homogeneous  conductor  considered.  See  XVI. 


216          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

A  general  characteristic  of  non-intrinsic  e.m.f.s  is  that  they 
have,  like  the  force  of  friction  in  mechanics  (analogous  to  the 
counter  e.m.f.  of  resistance,  —  RI\  or  like  the  elastic  reaction  of 
a  stretched  spring  (analogous  to  a  difference  of  potential),  no  in- 
dependent existence  of  their  own,  but  are  developed  only  through 
the  action  of  an  agent  possessing  a  true  intrinsic  electromotive 
force,  such  as  a  dynamo  or  a  voltaic  cell.  This  is  equivalent  to 
the  statement  that  electrical  energy  is  never  generated  at  the 
expense  of  energy  in  some  other  form  through  the  agency  of  a 
non-intrinsic  e.m.f. 

In  what  follows  many  cases  of  intrinsic  and  impressed  e.m.f.s 
will  be  considered. 

20.  Intrinsic  Electric  Intensity  and  E.M.F.  The  intrinsic  e.m.f. 
in  a  region  may  be  regarded  as  the  line  integral  of  an  intrinsic 
electric  intensity  in  the  region.  If  this  intensity  is  denoted  by  e, 
we  have  therefore 

V=fe-cos0-dL  (38) 

where  6  denotes  the  angle  between  the  direction  of  e  and  that  of 
the  element  of  the  path,  dL,  at  any  point. 

Consider  a  tube  of  flow  of  cross-section  dS  at  a  point  where 
the  intrinsic  intensity  is  e.  If  0  denotes  the  angle  between  the 
directions  of  e  and  of  /,  the  intrinsic  e.m.f.  between  two  right 
cross-sections  of  this  tube  distant  dL  apart  is  e  cos  6  •  dL.  The 
current  through  the  tube  is  i-dS.  Hence  the  rate  at  which 
power  is  transformed  into  the  circuit  per  unit  volume  at  the  point 

is 

dP  /dr  =  e-cos  O'dL-idS  jdSdL  =  ei'Cos  6  (39) 


From  this  power  equation  e  •  cos  6,  the  component  of  e  in  the 
direction  of  i,  might  be  defined  as 

*"cos  e  =  (dPj  dT)ji  (40) 

If  P  denotes  the  total  power  transformed  into  electrical  energy 
in  an  isolated   electric  circuit,   and   /  the   current,   then,    since 


THE    CONDUCTION    CURRENT.  2  1/ 


P  =  dW\dt  and  /=  dqjdt,  V  =  P/f  =  dWjdq.  That  is,  ¥  is 
the  work  per  unit  charge  done  in  carrying  a  charge  around  the 
circuit.  Hence  our  definition  of  an  intrinsic  e.m.f.,  and  therefore 
our  definition  of  an  intrinsic  intensity,  is  in  agreement  with  the 
general  definition  of  §  17,  I.  The  same  is  true  of  the  impressed 
e.m.f. 

When  P  is  expressed  in  ergs  per  second,  and  /  in  the  RES 
unit  current,  or  when  dWis  expressed  in  ergs  and  dq  in  the  RES 
unit  charge,  "^  is,  by  definition,  expressed  in  the  RES  unit  e.m.f. 

21,  Intrinsic  Electromotive  Forces  in  Series.  If  any  number, 
n,  of  agents  with  individual  electromotive  forces  Wv  ^2,  •  •  •  ,  M^ 
are  connected  up  in  series,  so  that  the  same  current  traverses 
each,  the  resultant  e.m.f.  is 

*-¥,  +  ¥,+  ...  +  *.  (39) 

proper  attention  being  paid  to  signs.  For  if  P,  Pv  P2,  etc.,  de- 
note the  power  supplied  to  the  circuit  by  the  resultant  e.m.f.  and 
the  powers  supplied  by  the  individual  e.m.f.s,  and  /the  current, 


from  which  (39)  immediately  follows. 

The  above  proposition,  demonstrated  for  intrinsic  e.m.f.s,  is 
obviously  also  true  for  the  more  general  impressed  e.m.f.s. 

Intrinsic  Electromotive  Forces  in  Multiple.  If  any  number,  n, 
of  similar  agents  having  the  same  e.m.f.  M*'  are  connected  up  in 
multiple,  so  that  one  /2th  of  the  current  traverses  each  in  the 
same  direction,  the  resultant  e.m.f.  "SP"  is  eqnal  to  W  '.  For 


l         2  n 

=  ¥'//»  +  ¥'//*  +  •  •  .  +  ¥'//»  =  ¥'/ 

22.  Ohm's  Law,  General  Form,  Deduced  from  Joule's  Law.  Let 
the  resistance  of  a  conductor  I  €2,  Fig.  72,  between  two  electrodes 
I  and  2  be  denoted  by  R.  Let  the  conductor  be  traversed  by  a 
current,  reckoned  positive  when  in  the  direction  I  €2,  with  the 


218          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

same  value  /  at  any  instant  across  every  section,  and  let  the  con- 
ductor be  acted  upon  by  an  impressed  e.m.f.  ^12  -f  M^/  =  P\I 
_j_  f»  jft  tyi2  and  W12  being  reckoned  positive  when  in  the  direc- 
tion of  the  current  I (i.  e.,  162).  "^12  is  the  intrinsic  e.m.f.  be- 
tween the  electrodes,  and  AJr12/  the  electromagnetic  energy  per 
unit  current  transferred  per  unit  time  to  the  region  I  €2  (negative 
when  power  is  transferred  from  I  62). 


Fig.  72. 

While  the  current  /  traverses  the  circuit,  the  conductor  I  €2 
receives  energy  at  the  rate 


and  its  resistance  dissipates  energy  in  heat  at  the  rate 

dHjdt  =  RI2 
Hence,  by  the  principle  of  the  conservation  of  energy, 

(40 


whence 

*„  +  *„'-*/  («) 

or  (42) 


(42)  (a)  states  that  the  impressed  e.m.f.  in  any  conductor  is 
equal  and  opposite  to  the  counter  e.m.f.  of  resistance  (  —  RI~). 

(42)  (£)  states  that  the  current  in  any  conductor  is  equal  to  the 
impressed  e.m.f.  acting  upon  the  conductor  divided  by  its  re- 
sistance. 

Either  of  these  statements  constitutes  Ohm's  law  in  its  general 
integral  form. 

20.  General  Differential  Form  of  Ohm's  Law.  Impressed  Elec- 
tric Intensity.  The  impressed  e.m.f.  in  a  region  may  be  regarded 


THE   CONDUCTION    CURRENT.  219 

as  the  line  integral  of  a  total  or  impressed  electric  intensity  E  = 
vector  sum  of  e  and  E  ,  the  field  intensity  F/g,  in  the  direction 
of  the  current  density  it  produces.  Thus 


.  (43) 

along  a  line  of  impressed  identity,  (e  +  -£')  being  a  vector  sum. 
Applying  (41)  and  (43)  to  a  stream-tube  whose  cross-section 
is  dS  at  a  point  /*  where  the  impressed  intensity  is  E=  e  -f  E  ', 
we  have,  for  the  element  of  volume  dT  =  dL'dS  enclosing  P  and 
bounded  by  the  sides  of  the  tube  and  two  right  cross-sections 
distant  dL  apart, 

EdL  -  idS  =  0  -f  Ef)dL  •  idS  =  rPdLdS  =  i^k  •  dLdS 
whence 

i=kE=  k(e  +  E')  =  E/r  =  (e  +  E')/r  (44) 

which  is  the  general  differential  form  of  Ohm's  law. 

When  E=  E  y  or  e  =  o,  (44)  reduces  to  (16). 

23.  Ohm's  Law  for  Constant  Current.  Let  the  electrodes  I 
and  2  of  the  conductor  I  £2,  Fig.  72,  be  maintained  at  the  poten- 
tial difference  F12  (with  the  assistance  of  an  intrinsic  e.m.f.  located 
outside  I  £2,  if  necessary)  while  the  conductor  I  €2  is  traversed 

tu 


by  a  constant  current  /.  Then  electromagnetic  energy  is  gen- 
erated in  the  conductor  at  the  rate  PI  =  '^r12/,  and  electromag- 
netic energy  is  transferred  from  the  field  into  the  conductor  at 
the  rate  P  =  ^I2ff==  F12/,  the  total  impressed  e.m.f.  in  the  con- 
ductor in  the  direction  I  €2  being  thus  FJ2  -J-  "*F12. 


220          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 
In  this  case,  therefore,  Ohm's  law  (42,  b)  becomes 


"-••'';       (45) 

(1)  If  F12  =  o,  that  is,  if  the  electrodes  I  and  2  are  connected 
together  as  in  Fig.  73,  to  form  a  closed  circuit,  (45)  becomes 

,,      f-VJR  •     ••-.•",  (46) 

which  is  Ohm's  law  for  a  closed  isolated  circuit  traversed  by  a 
constant  current. 

(2)  If  ^2  =  o,  that  is,  if  the  conductor  i  €2  is  homogeneous 
without  an  intrinsic  e.m.f.,  (45)  becomes 


which  is  Ohm's  law  for  a  homogeneous  conductor  traversed  by 
a  constant  current. 

(3)  If  J^2  -f  ^12  is  greater  than  zero,  /  has  the  direction  I  £2  ; 
if  V12  +  M/^  is  less  than  zero,  /  has  the  opposite  direction. 

(4)  If  F12  -f  ¥12  =  o,  that  is,  if  F|2  =  -  ¥12  =  ^21,  7=0,  and 
the  agent  with  intrinsic  e.m.f.  ^12  is  on  open  circuit.     Thus  the 
difference  of  potential  between  the  terminals  of  a  voltaic  cell,  or 
other  agent  possessing  an  intrinsic  e.m.f.  when  no  current  is  flow- 
ing, or  when  on  open  circuit,  is  equal  in  magnitude  to  the  intrinsic 
e.m.f.  for  zero  current,  but  has  the  opposite  direction. 

24.  Mechanical  Analogue  of  the  Relation  V12  +  ^12  =  El,  etc. 
Let  an  incompressible  liquid  flow  at  the  constant  rate  /  units 


Fig.  74. 

volume  per  second  in  the  direction  12  of  the  arrows,  Fig.  74, 
across  every  section  of  a  pipe  AB  containing  a  screw  propeller, 
or  pump,  C  producing  a  difference  of  pressure  "^12,  in  the  direc- 
tion of  the  current  (1^2),  on  its  two  sides.  If  the  sections  Pl 
and  P2  are  maintained  at  pressures  Fj  and  Vv  or  at  the  pressure 


THE  CONDUCTION  CURRENT.  221 

difference  F12  =  Vl  —  V^  the  total  fall  of  pressure  along  the 
pipe  from  Pl  to  P2  is  F12  +  ^12  =  +  Rf,  and  the  rate  at  which 
work  is  done  against  friction  within  the  volume  P±P2  is  (F12  + 
Wl2)f  =  RP,  R  having  the  significance  attached  to  it  in  §18. 

If  the  pipe  is  bent  around  and  P1  and  P2  joined  together,  so  as 
to  form  a  closed  circuit,  Vl=  V2  or  F12  =  o,  and  ^12  =  RI. 

If  elastic  membranes  are  stretched  across  the  pipe  at  P1  and  P2 
(either  before  or  after  Pl  and  P2  are  joined  together),  the  propeller 
will  force  liquid  from  the  region  A  into  the  region  B  (analogous 
to  the  positive  and  negative  charges  of  the  terminals  of  an  agent 
possessing  an  intrinsic  e.m.f,  when  on  open  circuit)  until  the 
pressure  in  B  exceeds  the  pressure  in  A  by  the  amount  V12  =  ^12, 
numerically,  when  the  current  will  cease. 

25.  The  Fall  of  Potential  Around  a  Closed  Circuit.  Consider 
a  closed  circuit  containing  an  agent  with  an  intrinsic  e.m.f.  M/" 
and  traversed  by  a  constant  current  /.  Let  the  resistance  of  the 
agent,  called  the  internal  resistance,  be  denoted  by  B,  and  that 
of  the  rest  of  the  circuit,  called  the  external  resistance,  by  Ry 
both  conductors  being  supposed  homogeneous.  Then  we  have, 
by  (46), 

Vb  +  Vr)  (48) 


Now  BI  denotes  the  fall  of  potential,  Vw  in  the  direction  of 
the  current  through  the  resistance  B  of  the  agent,  and  RI  the 
fall  of  potential,  F,  in  the  direction  of  the  current  through  the 
external  resistance  R.  But  the  total  fall  of  potential  around  a 
complete  circuit  is  zero  (§18,  I.).  Hence  at  the  seat  of  the  in- 
trinsic e.m.f.  there  is  a  rise  in  potential  in  the  direction  of  the 
current  equal  to  ¥  =  (B  +  R)I.  The  equation  -l\  -  Pj  •  JBf2,  Y/2* 
§  23,  (4),  is  a  particular  case  of  this  proposition  (R  =  infinity). 

To  make  the  fall  of  potential  as  great  as  possible  through  the 
external  circuit  it  is  clear  that  R  JB  should  be  made  as  great  as 
possible,  if  M*  is  independent  of  the  current. 

The  e.m.f.  of  an  agent  is,  in  general,  a  more  or  less  compli- 
cated function  of  the  current,  although  there  are  some  cases  in 


222          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

which  the  e.m.f.  is  constant  for  all  values  of  the  current.  The 
limiting  value  which  an  e.m.f.  approaches  as  the  current  ap- 
proaches zero,  and  the  resistance  infinity,  or  the  e.m.f.  on  open 
circuit,  is,  as  we  have  seen,  equal  and  opposite  to  the  potential 
difference  between  its  terminals  on  open  circuit.  If  V  denotes 
this  potential  difference,  and  if  ^  above  is  independent  of  the 
current,  we  have,  for  all  values  of  the  current, 

V=  Vb  +  Vr  (49) 

26.  Kirchhoff's  Law  II.  In  any  closed  circuit  in  a  network  of 
conductors  traversed  by  steady  currents,  as  the  circuit  I  2  3  4  •  •  •  n 
in  Fig.  75,  the  algebraic  sum  of  all  the  intrinsic  e.m.f.s  is  equal 


% 

Fig.  75. 

to  the  algebraic  sum  of  the  products  RI.  That  is,  if  N?12,  "^ 
.  •  -,  ^fnl  denote  the  intrinsic  e.m.f.s  in  the  branches  12,  23,  •  •  •  ;/i, 
Rlv  R^  .  •  • ,  Rnl  the  resistances  of  the  same  branches,  and  712,  723, 
. . .  ,  7Bl  the  currents,  both  currents  and  e.m.f.s  being  reckoned 
positive  in  the  same  direction,  as  12  •>  •  ni,  around  the  circuit, 

then 

£^  =  £  J?7  (50) 

For,  by  (45) 


from  which,  by  adding  up  both  members  separately,  we  obtain 
(So). 


THE   CONDUCTION   CURRENT. 


223 


27.  Wheatstone's  Bridge  consists  of  a  network  of  six  conduc- 
tors arranged  as  in  Fig.  76  or  Fig.  77,  with  an  intrinsic  e.m.f.  "¥ 
in  the  branch  13,  Fig.  76,  or  the  branch  24,  Fig.  77.  Let  the 
currents  in  the  branches  be  denoted  by  A,  B,  Cy  D,  Ft  and  G,  as 
shown  in  the  figure,  the  current  in  any  branch  being  positive 
when  in  the  direction  of  the  arrow-head  in  that  branch  ;  and  let 
the  corresponding  resistances  of  the  branches  be  denoted  by  a, 


bt  ct  d,f,  and  g.  First  we  shall  find,  from  Kirchhoff's  laws,  the 
current  G  in  the  branch  24  when  the  e.m.f.,  M*,  is  in  the  branch 
13,  Fig.  76. 

Applying  Kirchhoff's  law  I.  to  each  of  the  sets  of  conductors 
meeting  in  the  points  i,  2,  and  4,  we  obtain  the  relations 


C=A  +  G 
D=B-G 


(a) 


Applying  KirchhofPs   law  II.  to  each  of  the  closed  circuits 
1241,  2342,  and  1431,  we  obtain 

—  Gg  +  Aa—  Bb  =  o 
G    +  Cc  —  Dd=o 


On  eliminating   C,  D,  and  F  by  (a),  and  rearranging,  these 
equations  become 


224          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

—  gG  +  a  A  —  bB  =  o 
(g-  -f  c  +  d)G  +  cA  —  dB  =  o 

-  dG  +/A  +  (b  +d  +f)B  =  ¥ 
from  which 

(51) 


if  Q  is  written  for  the  determinant  of  the  coefficients  of  the  cur- 
rents in  the  last  set  of  equations,  viz., 


=  Q 

(52) 


-g  a  -b 

(g+c  +  d)  c  -d 

—  d  f         (b  +  d+f] 

The  current  in  any  other  branch  can  be  found  in  the  same 
manner. 

The  difference  of  potential  between  the  points  4  and  2  is 

(53) 


and  may  be  made  as  small  a  fraction  of  "SP  as  desired  by  giving  a 
suitable  value  to  (be  —  ad}. 

From  (41)  and  (52)  and  a  comparison  of  Figs.  76  and  77,  we 
see  that  when  the  e.m.f.  "SP"  is  in  the  branch  24,  Fig.  77,  the 
current  in  the  branch  13  is 

F=V(bc-ad)IQ  (54) 

Thus  the  current  in  the  conductor  24  due  to  e.m.f.  in  the 
conductor  1  3  is  equal  to  the  current  in  the  conductor  1  3  due  to 
the  same  e.m.f.  in  the  conductor  24,  all  the  resistances  remaining 
unaltered. 

By  a  similar  method,  this  reciprocal  relation  may  be  shown  to 
hold  for  any  two  branches  of  the  network,  or  any  network. 

When  be  =  ad,  the  current  in  either  of  the  two  conductors  1  3 
or  24,  due  to  an  e.m.f.  in  the  other,  is  zero.  The  two  conduc- 
tors are  then  said  to  be  conjugate.  In  this  case  the  conductor 
in  which  there  is  no  current  may  be  removed,  or  its  resistance 


THE    CONDUCTION    CURRENT.  225 

may  be  altered  in  any  manner,  without  affecting  the  state  of  the 
rest  of  the  system. 

From  the  relation  712  =  (V12+  ^l2}/^i2  and  the  principle  of 
superposition  of  potentials  and  e.m.f.s,  it  follows  immediately 
that  if  any  number  of  e.m.f.s  is  placed  in  the  network,  each  will 
produce  in  any  part  of  the  system  the  same  current  it  would  have 
produced  if  acting  alone.  The  current  in  any  branch  is  thus  the 
algebraic  sum  of  the  currents  due  to  each  e.m.f.  separately. 

Suppose  an  e.m.f.  placed  in  one  of  the  branches  12,  23,  34, 
or  41,  Fig.  76.  It  will  produce  a  current  in  the  other  branches, 
including  24.  But  if  be  =•  ad,  the  e.m.f.  in  13  will  produce  no 
current  in  24,  whatever  this  e.m.f.  may  be.  Suppose  the  e.m.f. 
^  to  have  such  a  magnitude  and  direction  as  to  produce  a  cur- 
rent in  the  branch  1 3  exactly  neutralising  the  current  in  the  same 
branch  due  to  the  e.m.f.  in  the  other  branch.  Then  there  is  no 
current  in  the  branch  13,  and  it  may  be  removed,  or  its  resistance 
may  be  made  infinite,  without  affecting  the  currents  in  the  other 
branches.  Thus,  when  be  =  ad,  the  current  in  24  is  independent 
of  the  resistance,  as  well  as  of  the  electromotive  force,  in  the 
branch  13.  This  result  is  applied  below  to  Mance's  method  of 
measuring  the  resistance  of  a  conductor  containing  an  intrinsic 
e.m.f. 

The  condition  that  the  two  conductors  1 3  and  24  may  be  con- 
jugate, viz.,  be  =  ad,  can  be  found  very  simply  as  follows.  Let 
the  voltages  between  the  points  12,  23,  34,  etc.,  Figs.  76  and 
77,  be  denoted  by  V12,  V&  VM,  etc.  Then  we  have,  as  the  con- 
dition that  G  may  be  zero  in  the  arrangement  of  Fig.  76,  V^  —  o. 
We  have  also,  in  this  case,  A  =  C,  and  B  =  D.  Therefore 

vl2  =  aA  =  VU  =  I>B 

and 

Dividing  aA  =  bB  by  cA  =  dB,  we  obtain  a  /  c  =  b  /  d,  or 

bc  =  ad  (55) 

UNIVERSITY  OF  CALIFORNIA 
ARTMENT  OF  CIVIL  ENGINEERS** 


226         ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

In  exactly  the  same  way  we  find  that  only  this  same  condi- 
tion must  be  satisfied  in  order  that  F  may  vanish  in  the  arrange- 
ment of  Fig.  77. 

Hence  the  conductors  13  and  24  are  conjugate  when  this  con- 
dition is  satisfied. 

The  Wheatstone's  bridge  is  very  extensively  applied  to  the 
comparison  of  electrical  resistances. 

28.  The  Comparison  of  Electrical  Resistances.     Thus  suppose 
we  have  an  unknown  resistance  a  which  is  to  be  compared  with 
a  standard  resistance  c.     The  resistances  a  and  c  are  connected  up 
with  two  other  resistances  b  and  d,  whose  ratio  must  be  known, 
as  in  Fig.  76  or  Fig.  77,  and  the  terminals  of  a  battery,  or  other 
agent  with  an  intrinsic  e.m.f.,  are  connected  to  the  points  I,  3  or 
2,  4,  and  an  electrometer  (or  galvanometer,  XII.)  to  the  points 
2,  4  or  1,3.     Then  the  resistance  c,  or  the  resistances  b  and  d,  or 
all  three,  are  varied  until  the  needle  of  the  electrometer  (or  gal- 
vanometer, which  is  almost  invariably  used)  remains  undeflected 
whether  the  branch  containing  the  battery  is  opened  or  closed. 
Then,  by  (5  5),  a  =  c  b  /  d. 

29.  Mance's    Method    of   Determining   the    Resistance    of   an 
Agent  with  an  Intrinsic  E.M.F.      The  agent  whose  resistance, 
a,  is  to  be  determined  is  connected  up  as  in  Fig.  76  with  three 
other  resistances  c,  b,  and  d,  at  least  one  of  which,  together  with 
the  ratio  of  the  other  two,  is  known.     A  galvanometer  or  elec- 
trometer G  is  connected  to  the  points  2,  4,  and  a  wire  containing 
a  key,  but  no  e.m.f.,  to  the  points  1,3.     Then  the  resistances  r,  b, 
and  d,  or  at  least  one  of  them,  are  varied  until  the  deflection  of 
the  galvanometer  or  electrometer  is  the  same  whether  the  key 
connecting  the  points  I   and   3   is  open  or  closed.     When  this 
condition  is  reached,  the  current  through  the  branch  24,  or  the 
voltage  V^  is  independent  of  the  resistance  of  the  conductor  13, 
and  the  two  conductors  are  conjugate.      Hence 

a  =  c  bid 


THE    CONDUCTION    CURRENT.  22/ 

30,  Kelvin's  Method  of  Measuring  the  Resistance  of  a  Galvano- 
meter or  other  Current  Indicator.     The  galvanometer  or  electrom- 
eter of  §  28  is  removed  and  is  replaced  by  a  wire  W  containing  a 
key,  and  the  instrument  whose  resistance  is  to  be  determined  is  put 
in  place  of  the  unknown  resistance  a.     Then  one  or  more  of  the 
resistances,  b,  c,  d,  are  varied  until,  the  battery  circuit  being  closed, 
the  permanent  indication  of  the  instrument  (a  deflection,  if  a  gal- 
vanometer or  similar  instrument  is  under  experiment ;    silence, 
if  a  telephone)  remains  constant  when  the  key  in  the  wire  W  is 
opened  or  closed.      Then  no  current  traverses  the  wire  in  either 
case,  and  a—  cbjd,  as  in  §  28. 

31.  Kelvin's  Double  Bridge  furnishes  the  most  accurate  known 
means  of  comparing  two  very  small  resistances.*     The  conductor 
A,  Fig.  76,  of  a  Wheatstone's  bridge  arranged  for  the  comparison 
of  resistances,  §  28,  is  disconnected  from  the  conductor  B  at  the 
point  I,  and  the  terminals  of  one  of  the  resistances  to  be  com- 
pared, x,  are  connected  to  the  free  end  of  A,  denoted  by  i',  and 
the  original  point  I.     In  like  manner,  £7  is  separated  from  D  and 
the  other  resistance   under  comparison,  yy  is  connected  to  the 
point  3  and  the  free  end  of  C,  denoted  by  3'.      The  bridge  is 
completed  by  joining  the  points  i'  and  3'  with  a  third  conductor 
of  low  resistance.     The  resistances  a,  b,  <r,  d  (all,  in  practise,  of 
considerable  magnitudes,  which  can  be  determined  with  precision 
by  other  methods),  or  either  c  and  d  or  a  and  b,  are  then  varied, 
the  ratio  bjd  being  kept  constantly  equal  to  a/c,  until  no  current 
traverses  the  galvanometer.     Then,  since  the  currents  through  a 
and  c  are  equal,  and  also  the  currents  through  b  and  d,  and  there- 
fore the  currents  through  x  and  yt  it  is  clear  that 

xly  =  ale  =  bjd  (56) 

For  a  thorough  discussion  of  the  Kelvin  bridge  in  its  general 
form,  see  Zeiischr.  fur  Instrumentenkunde ,  Feb.  and  Mar.,  1903. 
Equation  (56)  does  not  express  the  general  condition  for  a 
balance. 

*With  the  possible  exception  of  the  shunted  differential  galvanometer  method 
(F.  Kohlrausch,  Wied.  Ann.,  Vol.  20,  p.  76,  1883). 


CHAPTER   IX. 
ELECTROLYTIC  AND   METALLIC   CONDUCTION. 

1.  Metallic  Conduction.     The  electric  current  in  a  metallic  con- 
ductor, whether  a  pure  metal  or  an  alloy,  in  the  solid  or  liquid 
state,  is  not,  so  far  as   is  known,  associated  with  any  chemical 
change  in  the  conductor  or  with  the  convection  of  its  molecules 
or  atoms  from  one  part  to  another.     All  substances  which  con- 
duct in  this  manner  are  said  to  conduct  metallically.     A  theory 
of  metallic  conduction,  based  on  the  motion  of  electrons,  will  be 
referred  to  in  §15. 

2.  Electrolytic  Conduction.     Electrolysis.     Ions.     The  electric 
current  in  most  chemical  compounds,  however,  is  invariably  as- 
sociated with   their  separation  into  two  constituents,  atoms   or 
groups  of  atoms,  called  ions.     These  ions  do  not  appear  separ- 
ately in  the  body  of  the  conductor,  but  only  at  the  electrodes  by 
which  the  current  enters  and  leaves  it.      Hence  one  of  the  ions 
moves   toward   the    anode,  and  is    therefore  called   the  anion ; 
while  the  other  moves  toward  the  kathode,  and  is  called  the 
kation. 

Substances  in  which  the  electric  current  is  associated  with  the 
transportation  of  atoms  or  molecules  are  called  electrolytes,  the 
process  of  electro-separation  of  the  constituents  is  called  electroly- 
sis, and  the  substances  are  said  to  conduct  electrolytically . 

The  simplest  electrolytes,  in  some  respects,  are  molten  salts, 
e.  g.,  KC1  at  a  temperature  above  734°  C.  During  the  elec- 
trolysis of  this  salt,  K  appears  at  the  kathode  and  Cl  at  the 
anode.  Thus  K  is  the  kation  and  Cl  the  anion.  As  in  this 
case,  so  in  the  electrolysis  of  salts,  acids,  and  bases  generally, 
the  kation  invariably  consists  of  a  metal  or  hydrogen,  and  the 

228 


ELECTROLYTIC   AND    METALLIC    CONDUCTION.         229 

anion  of  an  acid  element  or  radical  (sometimes  combined  with  a 
metal). 

The  commonest  electrolytes  are  aqueous  and  other  solutions 
of  salts,  acids,  and  bases.  In  such  a  solution,  all  together  an 
electrolyte,  the  dissolved  substance,  not  the  solvent,  is  separated 
into  the  moving  ions.  Hence  the  dissolved  substance  itself  is 
often  spoken  of  as  the  electrolyte.  It  must  be  remarked,  how- 
ever, that  pure  dry  acids,  salts,  and  bases  at  ordinary  tempera- 
tures, as  well  as  pure  water  and  other  solvents  whose  solutions 
are  frequently  electrolytes,  are  either  not  conductors,  or  else 
possess  extremely  small  conductivities. 

The  actual  determination  of  the  constituents  forming  the  ions 
is  sometimes  a  matter  of  considerable  difficulty.  For  in  many 
cases  the  ions  do  not  themselves  separate  out  at  the  electrodes, 
but  on  reaching  the  electrodes,  combine  with  them  chemically, 
if  such  reaction  is  possible,  or  with  the  solvent,  if  the  first  reac- 
tion is  impossible.  If  neither  reaction  is  possible,  the  ions  col- 
lect at  the  electrodes  or  are  there  liberated. 

Thus  if  molten  KC1  is  electrolysed  with  a  platinum  anode  and 
a  kathode  of  graphite,  metallic  potassium  may  be  collected  at  the 
kathode,  but  at  the  anode  the  Cl  unites  with  the  platinum. 

Also,  if  an  aqueous  solution  of  H2SO4  is  electrolysed  between 
platinum  electrodes,  H  (the  kation)  appears  at  the  kathode, 
where  it  is  given  off  as  a  gas  (no  reaction  with  platinum  or  with 
water  being  possible)  ;  and  SO4  (the  anion)  appears  at  the  anode. 
But  SO4  can  neither  combine  with  platinum,  nor  can  it  exist  alone 
in  the  presence  of  water  ;  hence  it  reacts  with  the  latter  to  form 
H2SO4  and  O,  the  first  going  into  solution,  and  the  second  being 
evolved  as  a  gas. 

3.  The  Laws  of  Faraday.  Electrolytic  Measurement  of  Current. 
According  to  the  experiments  of  Faraday,  confirmed  by  all  later 
investigation, 

I.  The  mass  of  an  ion  deposited  on  an  electrode,  or  there  dis- 
solved, during  the  passage  of  a  current,  is  proportional  to  the 


230          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

electric  charge  crossing  the  electrode  during  the  deposit  or  solu- 
tion. Thus,  if  Ma  denotes  the  mass  of  an  ion  a  deposited  at  an 
electrode,  or  there  dissolved,  in  the  time  t,  while  the  charge  q 
crosses  the  electrode  (or  passes  through  the  electrolyte). 


where  Ka  =  Ma  jq  is  a  constant  for  the  ion  a,  equal  to  the  mass 
of  the  ion  deposited  per  unit  charge,  and  called  its  electrochemical 
equivalent. 

If  a  condenser  of  capacity  5  is  repeatedly  charged  to  a  voltage 
Fand  discharged  in  a  constant  direction  through  an  electrolyte 
at  the  rate  n  times  per  second,  the  mass  Ma  of  the  ion  a  deposited 
or  dissolved  in  the  time  t  will  be 

.-,,       '''"'"-^     -'^Mn  =  Knq  =  KaSVnt        •--;     '  (2) 

from  which,  by  measuring  MaJ  S,  V,  n,  and  t,  Ka  may  be  de- 
termined. If  Ma  is  expressed  in  grams,  t  in  seconds,  and  S  and 
Fin  RES  units,  Ka,  will  be  expressed  in  the  RES  unit  electro- 
chemical  equivalent. 

If  a  constant  Current  /traverses  the  electrolyte,  we  have 


KJt  (3) 

from  which 

I  -MM  (4) 

Hence  by  measuring  Ma,  t  and  Ka,  I  may  be  determined.  If 
Ma  and  t  are  measured  in  grams  and  seconds,  respectively,  and 
Ka  in  the  RES  unit,  /  will  be  expressed  in  the  RES  unit  current. 

II.  The  electrochemical  equivalent  of  any  ion  is  directly  pro- 
portional to  its  atomic  (or  combining)  weight  and  inversely  pro- 
portional to  its  valence  ;  i.  e.t  the  electrochemical  equivalent  of 
a  substance  is  proportional  to  its  chemical  equivalent.  Thus  if 
a  and  b  denote  two  ions,  A  and  B  their  atomic  (or  combining) 
weights,  a'  and  b'  their  valences,  and  Ka  and  Kb  their  electro- 
chemical equivalents,  then 


ELECTROLYTIC    AND    METALLIC    CONDUCTION.         231 

^tS^wd    K^  =  W      f^TS:f^& 
whence 

K^Aja'-b'IB-K,  (6) 

If  therefore  the  combining  weights  and  valences  of  all  ions  are 
known,  and  the  electrochemical  equivalent  of  any  one  of  them, 
the  electrochemical  equivalents  of  all  the  rest  may  be  found  from 
(6).  The  ion  whose  electrochemical  equivalent  has  been  most 
accurately  determined  is  the  silver  ion,  of  which  the  valence  is 
I,  the  combining  (atomic)  weight  107.93  (0  =  16.000),  and  the 
electrochemical  equivalent  0.001119  gram/coulomb  (XIV.). 
Hence  if  b-'m  (6)  denotes  silver,  the  electrochemical  equivalent 
of  any  other  ion  a  is 

Ka  =  A^'  •  1/107.93  •  x  o.ooi  1 19  gram/coulomb 
=  A/af  •  x  0.00001037  gram/coulomb 

By  omission,  law  II.  states  that  Ka  is  independent  of  the  nature 
of  the  compound  in  which  a  is  found,  the  nature  of  the  solvent 
if  the  compound  is  in  solution,  the  strength  of  the  current,  the 
temperature,  and  other  physical  conditions. 

If  an  element  has  two  or  more  valences  in  different  compounds, 
then  it  exists  as  two  or  more  distinct  ions,  each  with  its  own  elec- 
trochemical equivalent.  Thus  iron  in  ferric  compounds,  as  FeCl3, 
has  a  valence  3,  while  in  ferrous  compounds,  as  FeCl2,  its  valence 
is  2.  Hence 

.  .  valence  ferrous  iron 

K( ferric  iron)/K(terrous  iron)  = -. —    — ^ — ; — -. =4 

valence  ferric  iron 

since  the  atomic  weight  of  iron  is  the  same  for  both  classes  of 
compounds. 

Since  the  same  quantity  of  electric  charge  crosses  every  sec- 
tion of  a  conductor  carrying  a  steady  current  in  any  given  time, 
it  follows  from  law  II.  that  in  the  electrolysis  of  any  compound 
the  ions  are  deposited  simultaneously  at  the  two  electrodes  in  the 
same  proportions  in  which  they  occur  in  the  compound.  Thus 


2j2          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

when  KC1  is  electrolysed,  for  every  atom  of  K  deposited  at  the 
kathode,  an  atom  of  Cl  is  deposited  at  the  same  time  at  the 
anode.  Likewise,  in  the  electrolysis  of  an  aqueous  solution  of 
H2SO4,  for  every  ion  SO4  deposited  at  the  anode,  two  atoms  of 
H  are  deposited  at  the  same  time  upon  the  kathode.  Since  one 
sulphion  reacts  with  a  molecule  of  water  to  form  a  molecule  of 
H2SO4  and  an  atom  of  O,  it  follows  that  H  and  O  are  evolved  at 
the  kathode  and  anode  respectively  in  the  proportions  in  which 
they  occur  in  water,  while  the  total  quantity  of  H2SO4  in  solu- 
tion remains  constant.  In  like  manner,  when  an  aqueous  solu- 
tion of  silver  nitrate  is  electrolysed  between  silver  electrodes,  for 
every  silver  atom  (kation)  deposited  upon  the  kathode,  one 
nitrion  NO3  (anion)  is  liberated  at  the  silver  anode,  and  reacts 
with  an  atom  of  this  electrode  to  form  a  molecule  of  silver  nitrate. 
The  nitrate  goes  into  solution.  Thus  the  total  quantity  of  salt 
and  the  total  quantity  of  silver  in  solution  remain  constant,  while 
the  kathode  gains  as  much  silver  as  the  anode  loses. 

4.  The  Arrhenius  Theory  of  Electrolytic  Dissociation.  Ac- 
cording to  this  theory,  which,  though  not  universally  accepted, 
serves  to  explain  many  electrochemical  phenomena,  the  mole- 
cules of  a  molten  salt,  or  a  salt  in  aqueous  solution,  or  other 
electrolyte,  are  always,  in  greater  or  less  numbes,  independently 
of  the  passage  of  a  current,  broken  up,  or  dissociated,  each  into 
two  kinds  of  atoms  or  atomic  groups,  called  ions.  One  kind  of 
ion  is  positively  charged,  and  is  called  a  kation,  the  other  is  neg- 
atively charged  and  is  called  an  anion.  Thus  a  molecule  of 
H2SO4  dissociates  into  two  positively  charged  H  kations  and  one 
negatively  charged  SO4  anion  ;  and  a  molecule  of  KC1  into  a 
negative  chlorine  anion  and  a  positive  potassium  kation  the  metal- 
lic or  hydrogen  atoms  being  in  general  the  kations,  and  the  acid 
atoms  or  radicals  the  anions. 

Every  ion  of  the  same  valence  carries  a  charge  of  the  same 
magnitude,  and  this  charge  is  directly  proportional  to  the  va- 
lence of  the  ion.  Thus  the  negative  charge  of  a  chlorine  ion  is 


ELECTROLYTIC    AND    METALLIC    CONDUCTION.         233 

equal  to  the  positive  charge  of  a  potassium  ion,  the  valence  of 
each  being  I  ;  the  negative  charge  of  a  sulphion,  whose  valence 
is  2,  is  equal  to  the  positive  charge  of  two  hydrogen  ions,  whose 
valence  is  I  ;  the  charge  of  a  silver  ion  is  equal  to  the  charge  of 
a  nitrion,  and  to  the  charge  of  a  hydrogen,  potassium,  or  chlo- 
rine ion  ;  and  so  on. 

The  electric  conduction  current,  or  transfer  of  electric  charge, 
through  an  electrolyte  consists  in  a  convection  current  of  the 
ions  —  the  kations  with  their  positive  charges  moving  toward 
the  (negative)  kathode,  and  the  negative  anions  moving  toward 
the  (positive)  anode.  On  reaching  an  eiectode  the  ions  give  up 
their  charges  to  the  electrodes,  and  react  with  one  another,  or 
with  the  solvent,  or  with  the  electrode,  to  form  molecules. 

If  we  assume  that  the  current  at  each  electrode  consists  in 
the  motion  of  only  one  kind  of  ion,  which  will  presently  be 
proved  to  be  true,  the  theory  affords  a  simple  explanation  of  the 
laws  of  Faraday  : 

Since  a  given  kind  of  ion  always  carries  an  electric  charge  of 
the  same  magnitude  and  sign,  the  quantity  of  an  ion  deposited  upon 
an  electrode  will  be  proportional  to  the  charge  which  crosses  the 
electrode.  This  is  Faraday's  first  law. 

Since  the  charge  of  an  ion  is  proportional  to  its  valence,  and 
since  the  mass  of  an  ion  is  proportional  to  its  combining  weight, 
the  mass  of  ions  of  one  kind  carrying  a  given  charge,  or  the  ion 
mass  deposited  per  given  charge,  must  be  proportional  to  its 
combining  weight  and  inversely  proportional  to  its  valence. 
This  is  the  second  law  of  Faraday. 

5.  A  Gram  Atom  of  an  element  is  a  quantity  of  the  element 
equal,  in  grams,  to  the  number  denoting  its  atomic  weight.  Thus 
a  gram  atom  of  potassium  is  39. 1 5  grams  potassium. 

A  Gram  Ion  is  in  the  same  way  a  quantity  of  the  ion  equal,  in 
grams,  to  the  number  denoting  its  combining  weight.  Thus  a 
gram  ion  of  silver  is  107.93  grams  silver,  and  a  gram  sulphion 
is  96.06  =  (32.06  +  4  X  1 6.00)  grams  SO4. 


234         ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

A  gram  molecule  of  a  substance  is  a  quantity  of  the  substance 
equal,  in  grams,  to  the  number  denoting  its  molecular  weight. 
Thus  the  molecular  weight  of  KC1  is  39.15  +  35.45  =74.60, 
and  therefore  74.60  grams  KC1  is  a  gram  molecule  of  this  sub- 
stance. 

Tt  will  now  be  obvious  that  the  magnitude  of  the  charge  car- 
ried by  a  gram  ion  of  every  univalent  ion  is  the  same.  The 
charge  carried  by  a  gram  ion  of  a  bivalent  ion  is  twice  as  great, 
and  so  on.  The  charge  carried  by  a  gram  ion  is  equal,  numer- 
ically, to  the  gram  ion  divided  by  the  mass,  in  grams,  carrying 
unit  charge,  or  equal  to  the  gram  ion  divided  by  the  electro- 
chemical equivalent  of  the  ion.  Thus  the  silver  gram  ion  is 
107.93  grams,  and  the  electrochemical  equivalent  of  silver  is 
0.001119  gram  per  coulomb.  Hence  the  charge  carried  by  a 
gram  ion  of  silver  or  any  other  univalent  ion,  which  will  be  de- 
noted by  Q,  is 

Q  =  107.93  grams  -=-  o.ooi  1 19  gram/coulomb 

(8) 
=  96450  coulombs 

The  concentration  of  a  solution  is  the  quantity  of  dissolved 
substance  per  unit  volume  of  solution.  The  concentration  may 
be  expressed  in  grams  /c.c.,  grams /liter,  gram  molecules  /  liter, 
etc.,  and  will  be  denoted  here  by  C. 

6.  Velocities  of  the  Ions.  Hittorf  s  Ratio.  Hittorfs  Num- 
bers, etc.  If  in  the  electrolysis  of  a  solution  ordinary  convection 
and  diffusion  effects  are  prevented,  it  is  found  that  no  change 
takes  place  in  the  concentration  of  the  solution  except  in  the 
vicinity  of  the  electrodes.  Owing  to  the  deposit  or  liberation 
of  the  ions  at  the  electrodes,  however,  the  total  quantity  of  dis- 
solved substance  diminishes  (reactions  at  the  electrodes  which 
produce  the  substance  being  neglected).  Hence  it  follows  that 
the  concentration  of  the  dissolved  substance,  or  rather,  the  con- 
centration of  the  solution,  diminishes  near  the  electrodes  (the 
reactions  mentioned  being  neglected,  if  occurring). 


ELECTROLYTIC   AND    METALLIC    CONDUCTION.         235 

To  study  the  matter  more  closely  and  to  make  the  conditions 
perfectly  definite,  consider  the  electrolysis  of  an  aqueous  solu- 
tion of  silver  nitrate  between  platinum  electrodes,  A  and  K.  If 
we  imagine  a  porous  partition  P,  preventing  diffusion  and  con- 
vection currents  across  it  but  not  hindering  the  motion  of  the 
ions,  placed  between  A  and  K,  but  not  close  to  either,  the 
quantity  of  AgNO3  in  each  of  the  compartments  into  which  P 
divides  the  electrolytic  vessel  will  diminish  during  electrolysis. 
Before  electrolysis  begins,  each  compartment  contains  as  many 
anions  as  kations  (+  Ag  and  —  NO3),  one  of  each  being  neces- 
sary to  form  a  molecule  of  silver  nitrate.  During  electrolysis 
let  K  be  the  kathode  and  A  the  anode. 

Let  the  velocities  of  the  kations  and  anions  in  the  main  body 
of  the  solution,  at  the  partition  for  example,  be  denoted  by  U 
and  /^respectively,  and  let  Uj  V=  h.  Then  for  every  h  kations 
which  cross  the  partition  in  the  direction  AK,  i  anion  crosses  in 
the  direction  KA.  During  this  process  the  number  of  molecules 
of  AgNO3  in  the  compartment  KP  is  diminished  by  //,  since  both 
ions  are  necessary  to  form  a  molecule ;  and  likewise  the  number 
of  molecules  of  the  salt  in  the  compartment  AP  is  diminished  by 
i.  Hence  (Hittorfs  law) 

Loss  of  salt  near  anode /Loss  of  salt  near  kathode  =  Uj  V=  h  (9) 

The  ratio  k=  UjV  is  called  Hittorfs  ratio,  and  will  be  further 
discussed  below. 

When  h  kations  have  crossed  the  partition  in  the  direction  AK, 
and  one  anion  therefore  in  the  direction  KA,  h  -f  I  anions  are 
left  without  the  corresponding  kations  at  the  anode  A,  and  h  -f-  i 
kations  are  left  without  the  corresponding  anions  at  the  kathode 
K.  The  free  kations  immediately  give  up  their  positive  charges 
to  the  kathode,  and  are  deposited  upon  it ;  and  the  free  anions 
immediately  give  up  their  negative  charges  to  the  anode,  and 
react  with  water  to  form  nitric  acid  and  oxygen. 

Since  each  ion  carries  the  same  numerical  charge,  the  electric 
current  in  the  main  body  of  the  electrolyte  is  the  same  as  if  all 


236          ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

the  anions  were  moving  with  the  velocity  U+  V  to  ward  A,  and  all 
the  kations  were  at  rest  ;  or  as  if  all  the  kations  were  moving  with 
the  same  velocity  U  -\-  F  toward  K,  and  all  the  anions  were  at  rest. 

Since,  moreover,  by  what  precedes,  h  -f  I  kations  give  up  their 
charges  to  the  kathode,  and  h  4-  I  anions  give  up  their  charges 
to  the  anode,  while  only  h  kations  cross  the  partition  in  the  direc- 
tion AK  and  only  I  anion  in  the  direction  KA,  and  since  the 
current  is  the  same  across  every  section  of  the  conductor,  the 
current  at  the  anode  consists  in  the  motion  of  anions  only,  and 
the  current  at  the  kathode  consists  in  the  motion  of  kations  only. 
It  does  not  follow,  however,  that  the  velocity  of  either  anion  or 
kation  at  an  electrode  is  [/+  V.  For  the  quantity  of  kations 
deposited  at  the  kathode  is  greater  than  the  quantity  crossing  the 
partition  toward  the  kathode  in  the  ratio  (It  +  i)//i  =  (U  +  V)j  U\ 
and  the  quantity  of  anions  deposited  on  the  anode  is  greater  than 
the  quantity  crossing  the  partition  toward  the  anode  in  the  same 
time  in  the  ratio  (h  +  i)/  i  =  (U  +  V)  j  V. 

Thus  a  fraction  U  I  (U  +  F)  of  the  total  quantity  of  the  kation 
deposited  at  the  kathode  in  any  interval  comes  from  the  main 
body  of  the  electrolyte,  and  a  fraction  i  —  U  j(U  +  V)  = 
Vj(U  +  V}  comes  from  the  vicinity  of  the  kathode.  In  like 
manner,  a  fraction  V  I  (  U  -\-  F)  of  the  total  quantity  of  the  anion 
deposited  at  the  anode  in  any  interval  comes  from  the  main  body 
of  the  electrolyte,  and  a  fraction  i  —  V  j(U  +  F)  =  U  J  (U  +  V) 
comes  from  the  vicinity  of  the  anode. 

The  velocity,  Uv  of  the  kations  at  or  very  near  the  kathode, 
and  the  velocity,  Fj,  of  the  anions  at  the  anode,  can  be  obtained 
from  the  condition  that  the  charge  crossing  every  section  of  the 
conductor  in  the  same  interval  is  the  same.  This  condition  gives, 
for  the  interval  in  which  N  anions,  and  therefore  hN  kations,  cross 
the  partition, 


whence 

F)  =  [(^2  + 

i      U 


ELECTROLYTIC    AND    METALLIC    CONDUCTION.         237 
and 


+  i)  =  NV+  hNU 
whence 

)=U,  (10) 


The  ratios  Uj(U+  V)  and  V/(C7+  V\  which  represent  the 
fractions  of  the  total  current  carried  in  the  main  body  of  the  elec- 
trolyte by  the  kations  and  anions  respectively,  are  called  the 
transport  numbers  of  the  kation  and  anion,  respectively,  for  the 
given  electrolyte.  Put 

UI(U+V)  =  n  (ii) 

then 

F/(£7+F)=i-»  (12) 

and 

UIV-k-nl(i-n)  (13) 

If  the  electrolysis  of  silver  is  carried  on  between  silver  elec- 
trodes instead  of  platinum  electrodes,  the  loss  of  the  salt  around 
the  kathode  will  be  the  same,  for  a  given  charge,  as  before  ;  but 
since  the  total  quantity  of  salt  in  solution  now  remains  constant, 
there  will  be  a  gain  of  salt  around  the  anode  equal  to  the  loss 
around  the  kathode.  If  from  the  amount  of  AgNO3  correspond- 
ing to  the  total  quantity  of  silver  deposited  on  the  kathode  (or 
dissolved  at  the  anode),  which  would  be  the  gain  at  the  anode  if 
the  silver  ions  did  not  move,  we  subtract  the  actual  gain  at  the 
anode,  we  obtain  the  loss  of  salt  at  the  anode  due  to  the  motion 
of  the  silver  ions.  The  actual  gain  in  salt  at  the  anode  (equal 
to  the  loss  at  the  kathode)  divided  by  this  quantity  is  Hittorf's 
ratio,  from  which  n  and  I  —  n  are  easily  computed.  Since  any 
quantity  of  the  salt  is  proportional  to  the  quantity  of  silver  in  the 
salt,  we  may  use  the  quantities  of  silver  deposited  on  the  kathode 
and  gained  by  the  solution  around  the  anode  instead  of  the  cor- 
responding quantities  of  salt,  and  much  more  conveniently. 
Also,  the  loss  at  the  kathode  instead  of  the  gain  at  the  anode 
may  be  obtained,  if  preferable,  by  direct  experiment. 

Hittorf's  ratio,  and  therefore  the  transport  numbers  n  and  I  —  n, 
are  found  to  vary  slightly  with  the  temperature,  h  and  n  increas- 


238          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

ing  as  the  temperature  rises.  Thus,  for  NaCl,  n  =  0.392  at  20° 
C.  and  0.449  at  95°  C.;  and  for  AgNO3,  n  —  0.470  at  10°  C. 
and  0.490  at  90°  C. 

For  certain  electrolytes,  especially  aqueous  solutions  of  alka- 
line salts,  n  is  almost  independent  of  the  concentration.  Thus, 
in  the  case  of  an  aqueous  solution  of  KC1,  n  changes  from  0.497 
to  0.486  when  the  concentration  increases  from  0.03  to  2.5  grm. 
mol./lit. 

For  other  electrolytes  n  decreases  rapidly  with  the  concentra- 
tion. Thus,  for  an  aqueous  solution  of  CuSO4  n  decreases  from 
0.36  to  0.27  as  the  concentration  increases  from  o.i  to  2.0  grm. 
mol.  per  liter. 

For  still  other  electrolytes  ;/  increases  rapidly  with  the  increase 
of  concentration.  Thus,  for  an  aqueous  solution  of  AgNO3,  n 
changes  from  0.474  to  o.  5  3  as  the  concentration  increases  from 
o.oi  to  2  grm.  mol.  /liter. 

In  all  cases  n  is  independent  of  the  current  strength. 

7.  The  Dissociation  Ratio,     By  several  methods,  into  a  discus- 
sion of  which  we  cannot  here  enter,  for  example  the  lowering  of 
the  freezing  point  produced  by  the  solution  of  a  substance,  the 
ratio  of  the  number  of  dissociated  molecules  in  a  solution  to  the 
total  number  of  molecules  of  the  dissolved  substance  can,  ac- 
cording to  the  modern  dissociation  theory,  be  determined  with- 
out the  use  of  an  electric  current.     This  ratio,  called  the  disso- 
ciation ratio,  will  be  denoted  by  m.     For  a  given  solution  m  in 
general  increases  slowly  with  the  temperature. 

If  the  concentration  of  a  given  kind  of  electrolytic  solution 
is  diminished,  m  in  general  increases,  very  rapidly  at  first,  then 
more  and  more  slowly,  reaching,  when  the  solution  becomes  very 
dilute,  sensibly  the  constant  value  I.  That  is,  in  very  dilute 
solutions  all  the  molecules  of  a  dissolved  electrolytic  substance 
are  dissociated. 

8.  Ohm's  Law  for  a  Homogeneous  Electrolyte.     Conductivity 
and  Molecular  Conductivity.     Since  according  to  the  dissociation 


ELECTROLYTIC   AND    METALLIC    CONDUCTION.         239 

theory  the  electric  current  in  an  electrolyte  consists  in  the  motion 
of  the  ions,  the  undissociated  molecules  playing  no  part  in  the 
conduction,  the  current  density,  2,  must  be  equal  to  the  continued 
product  of  the  concentration  of  the  salt,  C,  In  gram  molecules 
per  cc.,  the  dissociation  ratio,  m,  the  valence  of  the  ions,  a' ' ,  the 
quantity  of  charge  carried  by  a  univalent  gram  ion,  Q,  and  the 
sum  of  the  ionic  velocities,  U  -\-  V.  For  mC  is  the  number  of 
gram  ions  of  each  kind  per  cc.,  mCa1  Q  is  the  charge  upon  all 
the  kations  in  one  cc.,  and,  numerically,  the  charge  upon  all  the 
anions  in  one  cc.;  hence  mCa'QU and  mCa' QV  are  the  total 
positive  and  negative  charges,  respectively,  crossing  unit  area  per 
second  in  opposite  directions.  Hence 

i=mCa'Q(U+V)  (14) 

Now  Ohm's  law  holds  rigorously  for  liquid  electrolytes  (though 
not,  in  general,  for  gases),  as  well  as  for  metallic  conductors. 
Hence  the  above  equation  may  be  written 

i=  kE  =  mCa'Q(U+  V)  (15) 

Therefore  the  sum  of  the  velocities  of  the  ions,  U  +  Vy  is  pro- 
portional to  the  electric  intensity  E.  Hence,  since  «  or  h  is 
independent  of  the  electric  intensity  or  current  (§6),  the  velocity 
of  each  ion  must  be  proportional  to  E.  If  therefore  u  and  v 
denote  the  velocities  of  the  kation  and  anion  respectively  per  unit 
intensity,  we  have,  when  the  intensity  is  E 

U=uE     and      V=vE  (16) 

(15)  may  therefore  be  written 

i  =  kE  =  mCa'Q(u  +  v)  E  (i» 

so  that 

k  =  mCa'Q(u  +  v)  (18) 

The  ratio  of  the  conductivity  k  to  the  concentration  C  (in  grm. 
mol./cc.)  is  called  the  molecular  conductivity,  and  will  be  denoted 
by  M.  Thus 

M=  kjC  =  ma'Q(u  +  v)  (19) 


240          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

When  the  solution  becomes  very  dilute,  ;//  becomes  sensibly 
equal  to  unity ;  hence,  if  the  corresponding  values  of  M,  ky  C,  it, 
and  v  are  denoted  by  these  letters  with  the  subscript  zero,  we 
have  from  (19) 

^  =  w="'fiK  +  o  (20) 

From  the  last  two  equations 

m  =  M/MQ-(uQ  +  v0)/(u  +  v)  (21) 

The  conductivity,  k,  of  an  electrolyte  is  readily  measured  by 
methods  similar  to  those  used  in  the  case  of  metallic  conductors, 
except  that  to  avoid  the  troublesome  effects  of  polarisation  an 
alternating  current  and  a  telephone  or  electrodynamometer  are 
employed  instead  of  a  direct  current  and  a  galvanometer.  From 
the  conductivity  and  the  concentration  the  molecular  conductivity 
is  obtained  by  division  from  (19).  From  the  molecular  conduc- 
tivity My  at  given  dilution,  and  MQt  m  can  be  computed,  if 

(V+ *•)/(*+*) 

is  known. 

9.  Variation  of  Electrolytic  Conductivity  with  Temperature 
Pressure,  and  Viscosity.  The  conductivity  of  an  electrolytic 
solution  always  increases  rapidly  with  the  temperature.  Since 
in  equation  (18)  a'  and  <2  are  constants,  and  mC  does  not  vary 
much  with  the  temperature,  this  increase  in  conductivity  must  be 
almost  wholly  due  to  an  increase  in  (u  -f  v).  Such  an  increase 
in  the  velocities  would  be  expected  from  the  fact  that  the  vis- 
cosity of  a  liquid  rapidly  diminishes  with  the  increase  of  tempera- 
ture. 

The  conductivity  of  an  electrolytic  solution  also  increases  with 
the  pressure  to  which  it  is  subjected.  Since  this  increase  occurs 
in  the  case  of  very  dilute  solutions  (m  —  i),  although  to  a  less 
degree  than  for  strong  solutions,  it  must  be  due,  in  part  at  least, 
to  the  diminutions  of  the  liquid's  viscosity  (which  always  dimin- 
ishes with  increase  of  pressure)  and  the  consequent  increase  of 
the  ionic  velocities. 


ELECTROLYTIC    AND    METALLIC    CONDUCTION.         241 


For  solutions  in  which  different  solvents  contain  the  same 
number  of  gram  ions  of  a  given  substance  per  cc.,  the  conductiv- 
ity is  always  less  the  greater  the  viscosity  of  the  solvent,  the 
conductivity  of  the  solvent  being  eliminated  if  appreciable. 

The  intimate  connection  between  the  ionic  velocities  and  vis- 
cosity is  shown  further  by  the  fact  that  the  conductivities  of  anal- 
ogous compounds  in  solution  bear  to  one  another  the  same 
ratios  as  do  their  velocities  of  diffusion. 

10.  Molecular  Conductivity  and  Concentration.  The  Dissocia- 
tion Ratio.  The  relation  between  the  molecular  conductivity  M 
of  an  aqueous  solution  of  KC1  and  the  dilution  of  the  solution, 
expressed  in  terms  of  the  number  of  liters  of  water  containing  a 


i.oo 


0.95 


0.90 


0.85 


20 


eo 


80 


Dilutior)(Liters  of  Water  Containing  One  Gram  Molecule  KCi.l 


Fig.  78. 

gram  molecule  of  the  salt,  is  shown  graphically  in  Fig.  78.  As 
the  dilution  increases,  or  the  concentration  diminishes,  M  in- 
creases, rapidly  at  first,  then  more  and  more  slowly,  reaching  a 
constant  value  MQ  when  the  solution  becomes  very  dilute. 

Curve  /of  Fig.  78  shows  the  relation  between  M/MQ  and  the 
dilution,  and  curve  //  the  relation  between  the  dissociation  ratio 
m,  calculated  from  the  lowering  of  the  freezing  point,  and  the 
dilution,  for  the  same  salt  KC1  in  aqueous  solution. 


242          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

The  close  similarity  of  the  curves  shows  that,  very  approxi- 
mately, in  this  case 

m  =  M;'MQ  (22) 

Hence,  to  the  same  degree  of  approximation,  equation  (21)  gives 
(11^  _j_  z/0)  =  (u  -f  v) ;  that  is,  the  sum  of  the  velocities  of  the  ions 
is  nearly  independent  of  the  concentration.  In  practise  this  ap- 
proximation is  frequently  employed,  and  the  dissociation  ratio  is 
calculated  from  (22)  on  the  assumption  that  (UQ  -f  ^0)  =  (u  -f  v) 
sensibly.  Equation  (22),  however,  appears  in  many  cases  to  be 
not  even  approximately  true  except  for  very  great  dilutions. 

11.  The  Velocities  of  the  Ions.    The  Law  of  Kohlrausch.     From 

(19)  it  follows  that 

(u  +  v)  =  M/ma'Q  (23) 

and  from  (9)  and  (16)  that 

h  =  Ul  V=  uEjvE  =  u/v  (24) 

From  these  equations  we  have 

u  =  hMJma'  Q  cm.  per  second  per  unit  intensity         (25) 
and 

v  —  (i  —  H)  Mjma'  Q  cm.  per  second  per  unit  intensity   (26) 

from  which  u  and  v  can  be  readily  computed,  all  the  quantities 
in  the  second  members  being  capable  of  experimental  determina- 
tion. 

The  ionic  velocities,  when  calculated  by  these  formulae,  for 
extremely  dilute  solutions  are  found  to  be  wholly  independent 
of  the  compounds  in  which  they  occur.  This  is  the  law  of  Kohl- 
rausch. Thus  the  velocity  of  the  chlorine  ion  is  the  same 
whether  in  a  very  dilute  solution  of  HC1  or  a  very  dilute  solu- 
tion of  NaCl.  The  velocities  of  the  hydrogen,  silver,  hydroxyl 
(OH),  and  chlorine  ions,  all  in  cms.  per  second,  when  the  elec- 
tric intensity  is  one  volt  per  cm.,  are  0.00320,  0.00057.  0.00181, 
and  0.00069,  respectively,  by  calculation  from  (25)  and  (26). 


ELECTROLYTIC  AND  METALLIC  CONDUCTION. 


243 


The  above  results  have  been  confirmed  by  the  direct  experi- 
ments of  Lodge  and  of  Whetham  (Philosophical  Transactions,  A, 
1893,  and  A,  1895). 

12.  Thermal  Analogue  of  Ohm's  Law.     Let  the  two  parallel 
faces,  i  and  2,  distant  L  apart,  of  a  large  plane  slab  of  a  homo- 
geneous isotropic  substance  with  thermal  conductivity  k'  be  main- 
tained at  the  temperatures  ^  and  t¥     Near  the  center  of  the 
slab  the  flow  of  heat  from  I  to   2  is  perpendicular  to  the  faces, 
and  the  fall  of  temperature  per  unit  length,  or  temperature  gradi- 
ent, from  i  to  2  is  uniform  and  equal  to  (^  —  t^)/L  =  E'  .     The 
time  rate  per  unit  area,  i'  ,  at  which  heat  crosses  a  plane  surface 
within  the  slab  parallel  to  its  faces  is 

i'  =  k'  E' 

which  is  strictly  analogous  to  Ohm's  law. 

13.  The  Variation  of  Metallic  Conductivity  with  Temperature. 
The  resistivity  of  all  pure  metals  increases  with  the  temperature, 
the  relation  between  the  resistivity  and  temperature  being  approxi- 
mately linear  for  ordinary  temperatures  according  to  the  equation 


The  coefficient  a  is  called  the  resistance  temperature  coefficient. 
For  many  metals  a  is  approximately  equal  to  1/273,  the  tem- 
perature coefficient  of  the  expansion  of  a  gas  at  constant  pressure, 
when  /0  is  taken  as  o°  C.  or  273°  absolute.  For  such  a  sub- 
stance we  have  approximately  from  the  above  formula 


where  t  is  the  absolute  temperature  at  which  the  resistivity  is  rt, 
and  r273  is  the  resistivity  at  the  temperature  273°  absolute. 

14.  The  Law  of  Wiedemann  and  Franz.  Not  only  is  the  law 
of  Ohm  analogous  to  the  law  of  thermal  conductivity,  but  for 
nearly  all  metals  and  alloys  the  ratio  of  the  thermal  conductivity 
k'  to  the  electrical  conductivity  k  is  approximately  the  same  at  a 


244         ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

given  temperature,  and  is  proportional  to  the  absolute  tempera- 
ture. This  indicates  that  the  processes  involved  in  the  two  kinds 
of  conduction  are  largely  identical. 


15,  The  Electron  Theory  of  Conduction.  According  toAthe 
comprehensive  and  rapidly  developing  electron  theory,  an  atom 
is  constituted  of  a  multitude  of  minute  particles,  called  electrons, 
each  carrying  a  permanent  and  constituent  electric  charge,  whose 
magnitude  is  that  of  the  charge  carried  by  a  univalent  ion  in 
the  electrolysis  of  liquids.  In  a  neutral  atom,  the  number  of 
positive  particles  is  equal  to  the  number  of  negative  particles  ; 
in  a  charged  atom  or  radical,  the  number  of  positive  electrons 
exceeds  the  number  of  negative  electrons,  or  vice  versa,  by  I,  2, 
3,  etc.,  according  as  the  ion  is  univalent,  bivalent,  trivalent,  etc. 
The  charge  of  a  single  electron  is  the  smallest  electric  charge 
which  can  exist,  and  no  charges  exist  except  the  charges  of 
electrons. 

In  the  electrolytic  conduction  of  liquids  no  free  electrons  take 
part,  but  the  current  consists,  as  we  have  already  seen  according 
to  the  dissociation  theory,  in  the  convection  of  the  ions,  all  of 
atomic  magnitude. 

In  electric  conduction  through  gases,  which  is  also  electro- 
lytic, the  positive  ions  are  atomic  in  magnitude  (atoms  or  radi- 
cals plus  or  minus  one  or  more  electrons  of  the  same  sign),  but 
the  negative  ions  are  frequently  single  electrons,  though  either 
may  be  loaded  down  with  an  agglomeration  of  neutral  mole- 
cules. The  electric  current  consists  in  the  convection  in  oppo- 
site directions  of  these  ions.  The  kathode  rays,  emitted  from 
the  kathode  in  a  highly  exhausted  vacuum  tube,  consist  of  nega- 
tive electrons  only,  moving  with  velocities  of  the  same  order  as 
that  of  light. 

In  metallic  conduction  the  main  body  of  the  metal  does  not 
participate  in  the  conduction.  The  current  consists  in  the  con- 
vection of  (temporarily)  free  positive  electrons  in  the  direction 
of  the  current,  or  in  the  convection  of  (temporarily)  free  nega- 


ELECTROLYTIC   AND    METALLIC    CONDUCTION.          245 

tive  electrons  in  the  opposite  direction,  or  in  both  processes 
simultaneously.  The  number  of  electrons  taking  part  in  metal- 
lic conduction  is  small  in  comparison  with  the  number  taking 
part  in  electrolytic  conduction  proper,  and  the  velocity  is  rela- 
tively great 

Thermal  conduction  also  takes  place  by  the  motion  of  elec- 
trons, but  under  a  temperature  gradient  both  positive  and  nega- 
tive electrons  move  in  the  same  direction. 

For  an  extended  discussion  of  the  electron  theory  as  applied 
to  electric  and  thermal  conduction  in  metals,  reference  must  be 
made  to  memoirs  by  Drude,  Ann.  der  Physik,  I.,  p.  566,  1900, 
III.,  p.  369,  1900,  and  VII.,  p.  687,  1902.  For  a  general  treat- 
ment of  the  electron  theory,  see  also  Lord  Kelvin,  Philosophical 
Magazine  (6),  III.,  p.  257,  1902,  and  Larmor's  ALther  and  Mat- 
ter. For  a  sketch  of  the  development  of  the  electron  idea,  with 
abundant  references,  see  Kaufmann,  Physikalische  Zeitschrift, 
III.,  p.  9,  1901,  or  a  translation  of  the  same  in  The  Electrician, 
November  8,  1901.  An  elementary  treatment  of  the  subject  is 
given  by  Lodge  in  The  Electrician,  Vols.  50  and  51,  1902—1903. 


CHAPTER   X. 
THERMAL   AND    VOLTAIC    ELECTROMOTIVE   FORCES. 

1.  The  Law  of  Volta.     Around  a  circuit  made  up  of  any  num- 
ber of  different  metals  connected  end  to  end,  there  is  no  resultant 
e.m.f.,  and  therefore  no  electric  current,  when  all  parts  of  the 
circuit  are  at  the  same  temperature  (unless  there  is  a  changing 
magnetic  flux  through  the  circuit,  XIII.). 

2.  The  Seebeck  Effect,     Around  a  circuit  formed  of  two  dif- 
ferent metals  a  current  flows,  in  general,  when  the  two  junctions 
are  at  different  temperatures. 

Such  a  circuit  is  called  a  thermocouple,  or  a  thermoelement. 

In  a  thermocouple  consisting  of  a  copper  wire  and  an  iron 
wire,  if  the  mean  temperatute  of  the  junctions  is  less  than  275° 
C.,  a  current  flows  from  the  copper  to  the  iron  across  the  hot 
junction. 

3.  The  Peltier  Effect.     When  an  electric  current  flows  across 
the  junction  of  two  metals,  heat  is  there,  in  general,  either  ab- 
sorbed (thermal  energy  transformed  into  electrical  energy)  or 
emitted  (electrical  energy  transformed  into  heat),  according  to 
the  direction  of  the  current.     The  rate  of  the  energy  transfor- 
mation is  proportional  to  the  current  strength,  and  the  process  is 
completely  reversible. 

Thus  at  a  copper-iron  junction  heat  is  absorbed  when  the  cur- 
rent passes  from  Cu  to  Fe ;  and  heat  is  emitted  at  the  same  rate 
when  the  same  current  crosses  the  junction  (maintained  at  the 
same  temperature)  in  the  opposite  direction. 

The  energy  transformations  occurring  during  the  circulation  of 
the  current  in  the  thermoelement  of  copper  and  iron,  §  2,  thus 
tend  to  cool  the  hot  junction  and  to  heat  the  cold  junction. 

246 


THERMAL   AND    VOLTAIC    E.M.F.S.  247 

The  Peltier  E.M.F.  The  junction  of  two  metals  is  therefore 
the  seat  of  an  intrinsic  e.m.f.,  called  the  Peltier  e.m.f.,  which  is 
constant  at  a  given  temperature  of  the  junction  for  all  values  of 
the  current.  The  e.m.f.  varies  with  the  nature  of  the  metals  and 
with  the  temperature  of  the  junction.  At  the  junction  of  two 
metals  A  and  B,  at  the  temperature  t,  the  Peltier  e.m.f.  acting 
from  A  to  B  will  be  denoted  by  tPab.  See  §  I,  I. 

If  the  two  junctions  of  a  thermoelement  have  the  same  tem- 
perature ty  tPab,  the  e.m.f.  from  A  to  B,  will  have  the  same  value 
at  both  junctions.  Hence  no  current  will  traverse  the  circuit, 
but  a  difference  of  potential  will  be  developed,  B  coming  to  a 
potential  f^  higher  than  that  of  A. 

Since,  however,  tPab  is  a  function  of  the  temperature,  there  will, 
when  the  two  junctions  are  at  different  temperatures  /x  and  tv  be 
a  resultant  Peltier  e.m.f.  around  the  circuit,  equal,  when  mea- 
sured in  the  direction  around  the  circuit  from  A  to  B  across  the 
junction  at  temperature  tv  to 

tfab  +  tflm  =s  tfab  —  tfab  (0 

4.  The  Thomson  Effect.  When  an  electric  current  traverses  a 
conductor  along  which  there  is  a  temperature  gradient,  heat  is 
either  absorbed  (transformed  into  electrical  energy)  or  emitted 
(electrical  energy  transformed  into  heat),  according  to  the  direc- 
tion of  the  current,  throughout  the  portion  of  the  conductor  in 
which  the  temperature  gradient  exists.  The  rate  of  energy  trans- 
formation is  directly  proportional  to  the  strength  of  the  current, 
and  depends,  so  far  as  temperature  is  concerned,  only  on  the 
temperatures  of  the  ends  of  the  conductor.  The  energy  trans- 
formations are  perfectly  reversible,  changing  sign,  but  not  mag- 
nitude, with  the  direction  of  the  current. 

The  absorption  or  emission  of  heat  just  described  takes  place 
in  addition  to  the  evolution  of  heat  according  to  Joule's  law  at  a 
rate  proportional  to  the  square  of  the  current. 

The  Thomson  E.M.F.  A  conductor  in  which  there  is  a  tem- 
perature gradient  is  therefore  the  seat  of  an  intrinsic  e.m.f.  which 


248  ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

is  constant  for  all  values  of  the  current  for  given  temperatures 
of  its  terminals.  This  e.m.f.  is  called  the  Thomson  e.m.f.  in  the 
conductor,  and  is  considered  positive  when  it  is  directed  from 
the  lower  to  the  higher  temperature.  If  ^  and  t2  denote  the 
temperatures  of  the  cooler  and  hotter  ends  of  a  conductor  A,  the 
Thomson  e.m.f.  from  the  cooler  to  the  hotter  end  is  denoted  by 
T 

hh  *  a- 

The  fact  that  the  Thomson  e.m.f.  (or  the  corresponding  en- 
ergy transformations)  depend,  so  far  as  temperature  is  concerned, 
only  on  the  temperatures  of  the  ends  of  the  conductor,  follows 
from  the  law  of  Magnus,  which  states  that  in  a  circuit  com- 
posed of  a  single  homogeneous  metal  there  is  no  electric  current, 
howsoever  the  temperature  varies  from  point  to  point.  Thus  the 
Thomson  e.m.f.  from  a  point  A  to  another  point  B  of  the  circuit 
is  the  same  either  way  around  the  circuit.  (The  law  of  Magnus, 
and  the  deduction  therefrom,  do  not  hold  in  certain  extreme 
cases,  as  when  the  cross-section  of  the  conductor  changes  sud- 
denly, etc.;  also,  at  least  in  certain  cases,  when  a  portion  of  the 
circuit  is  magnetised,  when  it  is,  strictly,  non-homogeneous.) 

Let  ISadt  denote  the  rate  of  heat  absorption  in  the  elementary 
length  dL  of  a  conductor  A,  the  mean  temperature  within  the 
length  dL  being  /  and  the  rise  in  temperature  from  one  end  to 
the  other  being  dt,  when  the  current  7  flows  up  the  temperature 
gradient.  Then  the  Thomson  e.m.f.  up  the  temperature  gradient 

' 


vS  is,  in  general,  a  function  of  the  temperature  and  varies  from 
substance  to  substance.  If  the  temperatures  of  the  cooler  and 
hotter  ends  of  the  conductor  are  tl  and  tv  respectively,  we  have, 
on  integrating  (2)  from  one  end  of  the  conductor  to  the  other, 

i  (3) 

By  an  obvious  thermal  analogy,  Su  is  called  the  specific  heat 
of  electricity  in  the  metal  A  at  the  temperature  /. 


THERMAL   AND    VOLTAIC    E.M.F.S.  249 

In  certain  substances,  e.  g.  copper,  5  is  positive ;  that  is,  heat 
is  absorbed  when  the  current  flows  up  the  temperature  gradient. 
In  others,  as  iron,  5  is  negative  ;  that  is,  heat  is  absorbed  when 
the  current  flows,  down  the  temperature  gradient. 

In  a  copper-iron  thermoelement,  therefore,  with  junctions  at 
temperatures  /L  and  tv  the  Thomson  e.m.f.  in  each  metal  has  the 
same  direction  as  the  current  in  that  metal  —  up  the  gradient  in 
the  copper,  and  down  the  gradient  in  the  iron. 

If  Sa  and  Sb  denote  the  value  of  6"  at  the  temperature  /  for  two 
metals  A  and  B  forming  a  thermocouple  with  the  cold  and  hot 
junctions  at  temperatures  tl  and  tv  respectively,  the  total  Thom- 
son e.m.f.  around  the  circuit  in  the  direction  from  A  to  B  across 
the  hot  junction  is 

&h 

5.  The  Total  Thermal  Electromotive  Force  in  a  Circuit  consisting 
of  two  homogeneous  metals  is  the  sum  of  the  two  Peltier  e.m.f.s 
at  the  junctions  and  the  two  Thomson  e.m.f.s  along  the  con- 
ductors. Thus  if  "Vab  denotes  the  total  e.m.f.  in  the  circuit, 
measured  in  the  direction  around  the  circuit  from  A  to  B 
across  the  hot  junction, 

•\\f    p          p 

ab         to     ab         t\     ab 


6.  The  Law  of  Intermediate  Metals  (Becquerel's  Law  I.).     If  at 

one  of  the  junctions,  at  temperature  t,  of  two  metals  A  and  B 
forming  a  thermoelement  a  third  metal  C  is  inserted  between  A 
and  B,  and  if  the  two  resulting  junctions  are  kept  at  the  original 
temperature  /  of  the  junction  AB  before  the  insertion  of  C,  the 
total  e.m.f.  of  the  circuit  is  not  altered.  The  total  Thomson 
e.m.f.  in  C  is  evidently  zero,  since  its  two  ends  are  at  the  same 
temperature  ;  hence  the  law  states  that 

fab  =  fac  +  fcb  (6) 

Thus  two  wires  may  be  soldered  together,  instead  of  being 
welded  or  twisted,  without  affecting  the  e.m.f.  of  the  junction. 


250          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

§§  3,  4,  and  6  completely  account  for  the  law  of  Volta. 

7.  The  Law  of  Successive  Temperatures  (Becquerel's  Law  II.). 
Consider  a  thermoelement  of  two  metals  A  and  B  with  the  junc- 
tions at  temperatures  ^  and  tv  respectively.      Let  the  total  e.m.f. 
around  the  circuit  in  the  direction  from  A  to  B  across  the  junc- 
tion at  temperature  /2  be  denoted  by  tlt^ab.     Then,  with  similar 
nomenclature,  if  experiments  are  made  with  one  junction  of  the 
element  at  temperature  tl  and  the  other  at  /',  then  with  the  first 
at  temperature  tf  and  the  other  at  /",  and  so  on,  and  finally  with 
the  first  at  temperature  tn  and  the  other  at  temperature  /2,  it  will 
be  found  that,  whatever  the  values  of  »,  /',  tn  ',  •  •  •,  /n, 

^2      a&  =  tit'^ab  ~f~  t't't^ab  T~   *  *  "  T  tnt^ab  \7) 

8.  Thermoelectric  Power.     If  d^ab  denotes  the  total  thermal 
electromotive  force  around  the  circuit  of  a  thermoelement  AB  in 
the  direction  from  A  to  B  across  the  junction  whose  temperature 
is  /,  when  the  temperature  of  the  other  junction  is  /  —  dt,  the 
differential  coefficient  d^abjdt,  the  e.m.f.  per  unit  difference  of 
temperature,  is  called  the  thermoelectric  power  of  the  metal  A 
with  respect  to  the  metal  B  at  the  temperature  /,  or  the  thermo- 
electric power  of  the  thermoelement  AB  at  the  temperature  /,  and 
will  be  denoted  be  tpab.     Thus  we  have 


For  the  total  e.m.f.  in  the  circuit  in  terms  of  the  thermoelec- 
tric power  /,  we  have  from  (7)  and  (8) 

S/'  (9) 

9.  The  Thermoelectric  E.M.F.  of  a  Copper-Iron  Element  at 
Moderate  Temperatures.  The  relation  between  the  total  thermal 
electromotive  force  of  a  copper  iron  thermoelement  and  the  tem- 
perature t  of  one  of  the  junctions,  when  the  other  junction  is  kept 
at  the  constant  temperature  of  o°  C.,  is  shown  graphically  in 
Fig.  79,  for  temperatures  up  to  600°  C.  As  /  increases,  the 
e.m.f.  around  the  circuit  in  the  direction  from  copper  to  iron  at 


THERMAL   AND    VOLTAIC    E.M.F.S. 


251 


the  junction  whose  temperature  is  t  increases  from  a  nega- 
tive value,  for  t  less  than  o°  C,  to  a  maximum  positive  value 
when  /=  275°  C.  As  t  continues  to  increase,  the  e.m.f.  de- 
creases, falling  to  zero  at  t=  550°  C.  Beyond  this  temperature 
the  e.m.f.  is  negative,  as  when  t  was  less  than  o°  C.,  that  is,  the 
current  flows  (or  the  e.m.f.  is  directed)  from  iron  to  copper  across 
the  junction  at  temperature  /. 


3000 
"nnn 

/ 

>^ 

X 

\ 

2 

0 

^-000- 
u_ 

5 

LU 

/ 

/  ELECTR 
COPPER- 

DMOTIVE  FC 
RON  THERI 

)RCE  OF  A 
/IO  COUPLE 

\ 

/ 

\ 

-100             O/ 

1C 

0                  2( 

)0                   300                  4 

0                  5C 

0         \       6( 

)0 

-1-  i-ooo- 

Temperatu 

e  in  Degrees 

Centigrade 

\ 

\ 

Fig,  79. 

From  the  curve  in  Fig.  79,  the  thermoelectric  power  of  cop- 
per with  respect  to  iron,  ,/c.,  can  be  readily  obtained  for  all  tem- 
peratures within  the  limits  of  the  curve.  For  if  a  tangent  is 
drawn  from  the  point  of  the  curve  corresponding  to  any  tem- 
perature /,  the  tangent  of  the  angle  made  by  this  line  with  the 
axis  of  temperatures  is 


The  relation  between  tpri  and  /  is  given  in  the  curve  of  Fig. 
80,  which  is  a  straight  line.  That  is,  the  thermoelectric  power 
of  copper  with  respect  to  iron  is  a  linear  function  of  the  tempera- 

^ci  =  ^J^=Alf+A2  (10) 

where  Al  and^2  are  constants  for  the  given  element  (copper-iron). 


252 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


The  temperature  corresponding  to  the  point  B(t  =  275°  C.Y 
for  which  tpci  —  o,  is  called  the  neutral  temperature  for  copper 
and  iron.  The  temperature  corresponding  to  the  point  C 
(t  —  550°  C.),  in  passing  which  the  electromotive  force  around  the 
circuit  is  reversed  in  direction  when  one  of  the  junctions  is  at 
o°  C.,  is  called  the  temperature  of  inversion  of  copper  and  iron 
with  respect  to  the  temperature  o°  C.  of  the  cooler  junction. 


X 

| 

1 

o 

I 

i 

Q 

>v  J? 

THERMC 

X 

ELECTRIC 
VITH  RESPE 

3OWER  OF 
CT  TO  IRQ 

COPPER 

Nj. 

"o 
> 
2. 
u 

2 

A 

X 

T 

xg 

\ 

\ 

T' 

C 

100       'I 
0. 

0                   1 

)0                   2 
Temperati 

30            jN 

re  in  Degrees 

100                 4 
Centigrade 

30                    5 
If 

00                 6 

r 

~« 

1 

\ 

X, 

Fig.  80. 

The  total  e.m.f.  ^  of  the  couple  is  represented  in  this  figure 
by  the  area  included  between  the  curve  and  the  axis  of  tempera- 
tures, and  between  the  two  perpendiculars  dropped  from  the 
curve  to  the  points  on  this  axis  corresponding  to  the  tempera- 
tures of  the  junctions.  Thus,  if  the  junctions  are  at  temperatures 
Tand  A, 


In  like  manner, 

ABWc.  =  area  ABFA 

ATWci  =  area  ABFA  +  (negative)  area  BT'HB 

etc.     This  follows  directly  from  (9). 


THERMAL   AND    VOLTAIC    E.M.F.S. 


253 


The  curve  showing  the  relation  between  the  thermoelectric 
power  of  an  element  and  the  temperature  is  called  the  thermo- 
electric line  of  that  element.  The  thermoelectric  lines  of  nearly 
all  thermoelements  consisting  of  either  pure  metals  or  alloys  are 
straight  lines  over  a  considerable  range  of  temperature,  like  that 
of  the  copper-iron  couple,  Fig.  80. 

If  the  thermoelectric  line  of  a  given  element  is  a  straight  line, 
the  curve  showing  the  relation  between  the  total  thermal  electro- 
motive force  in  the  circuit  and  the  temperature  t  of  one  junction, 
the  temperature  ^  of  the  other  being  kept  constant,  is  a  parabola. 
For  we  have,  by  integration  of  (10)  between  the  limits  tl  and  t, 


ti 

which  is  the  equation  of  a  parabola  with  its  axis  in  the  negative 
direction  of  the  axis  of  e.m.f.s. 

10,  Becquerel's  Law  III.     At  a  given  temperature  the  thermo- 
electric power  of  a  metal  A  with  respect  to  a  metal  C  is  equal  to 


Temperature 


Fig.  81. 

the  thermoelectric  power  of  the  metal  A  with  respect  to  any 
other  metal  B  plus  the  thermoelectric  power  of  B  with  respect  to 
C.  That  is,  .  .  .  ,  /  _  0\ 

,Ac=*A&  +  <Ac  (I2) 

Hence  if  the  thermoelectric  lines  are  drawn  for  two  metals  A 
and  B  with  respect  to  the  same   metal  C  (Fig.  Si),  tpab  at   any 


254          ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

temperature  /  can  be  obtained  by  subtracting  from  the  ordinate 
tPac  the  ordinate  tpbc. 

The  total  e.m.f.  tlt^ab  is  given  by  the  area  EFGHE  of  the 
figure. 

P,  the  point  of  intersection  of  the  two  lines  A  and  B^  is  the 
neutral  point,  or  the  point  corresponding  to  the  neutral  tempera- 
ture, for  the  metals  A  and  B,  since  there 


11.  The  Thermoelectric  Circuit  Treated  as  a  Reversible  Thermo- 
dynamic  Engine.  So  far  as  the  Thomson  and  Peltier  effects  are 
concerned,  the  absorption  and  evolution  of  heat  in  a  thermoelec- 
tric circuit  are,  as  we  have  seen,  proportional  to  the  current 
strength  and  the  time,  or  to  the  total  charge  which  has  passed 
through  the  circuit,  and  completely  reversible,  changing  sign 
with  the  direction  of  the  current.  There  are  other  thermal  proc- 
esses going  on  in  the  circuit,  however,  which  are  not  reversible  : 
the  conduction  of  the  heat  from  the  hotter  to  the  cooler  junction, 
which  bears  no  direct  relation  to  the  electrical  phenomena  ;  the  evo- 
lution of  heat  according  to  Joule's  law  at  a  rate  proportional  to  the 
square  of  the  current  ;  and  the  radiation  of  heat,  which,  like  its 
conduction,  bears  no  direct  relation  to  the  electrical  phenomena. 
Since  by  diminishing  the  current  the  second  effect,  being  propor- 
tional to  the  square  of  the  current,  may  be  made  as  small  as  we 
please  in  comparison  with  the  Thomson  and  Peltier  effects,  which 
are  proportional  to  the  first  power  of  the  currents  ;  and  since  the 
first  and  third  effects  have  no  direct  relation  to  the  electrical 
phenomena  ;  we  shall  assume  that  the  total  Thomson  and  Pel- 
tier e.m.f.s  are  not  affected  by  these  irreversible  processes,  and 
that  the  relations  between  them  can  be  obtained  by  treating  the 
circuit  as  a  perfectly  reversible  thermodynamic  engine,  all  irre- 
versible effects  being  neglected.  The  application  in  this  manner 
of  the  principles  of  thermodynamics  to  the  matter  in  question  is 
justified  by  the  approximate  agreement  with  experiment  of  the 
results  to  which  it  leads. 


THERMAL   AND    VOLTAIC    E.M.F.S.  255 

The  first  law  of  thermodynamics,  or  the  principle  of  the  con- 
servation of  energy,  together  with  experiment,  has  furnished  us 
with  the  relation 

(13) 

If  we  apply  (13)  to  the  case  in  which  t^  =  t,  and  t2  =  t  +dty 
or  if  we  simply  differentiate  (13)  with  respect  to  /,  v/e  obtain 


Sa  -Sb  (14) 

The  second  law  of  thermodynamics  furnishes  another  relation. 
Let  dH  denote  the  quantity  of  heat  absorbed  into  the  circuit  at 
the  temperature  /,  while  a  charge  dq  (=  current  X  time)  is 
carried  around  the  circuit  once.  Then  we  have,  for  the  whole 
cycle,  by  the  second  law  of  thermodynamics, 

=  dq 


-  AM  +      f  Va  - 

•& 


the  temperature  being  expressed  on  the  absolute  scale. 

Applying  this  equation  to  the  case  in   which  tv  =  t  and  /2  = 
/  -f  dt,  or  simply  differentiating  the  equation  with  respect  to  /, 

we  obtain 

d(fJt)ldt+(Sa-Sb)lt=o  (16) 

(16)  may  be  written 

d(f^ldt  -  ,PJt  +  Sa  -  Sb  =  o  (17) 

The  combination  of  this  equation  with  (14)  gives 


or 

f^J'J=td^atldt  (19) 

The  combination  of  (16)  and  (19)  gives 

(20) 


256          ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

Since  at  the   neutral    temperature   for  the  metals  A  and  B 
.p^  =  o,  (19)  gives,  for  this  temperature, 


That  is,  when  one  junction  of  two  metals  is  at  the  neutral 
temperature,  there  is  at  this  junction  no  Peltier  e.m.f.  and  no 
absorption  or  evolution  of  heat,  /,  the  absolute  temperature,  being 
always  greater  than  zero. 

If  6ub  denotes  the  angle  made  with  the  axis  of  temperatures  by 
the  tangent  to  the  thermoelectric  line  of  A  with  respect  to  B  at 
the  point  on  the  line  corresponding  to  the  temperature  /,  then 
(18)  gives  •  (22) 


(20)  and  (22)  give 

s,-sb=-tte*eab  (23) 

In  the  common  case  in  which  ^p^  is  a  linear  function  of  the 
temperature,  or  the  thermoelectric  line  straight,  tan  9ab  is  constant 
(Kab)  for  all  temperatures,  and  (23)  becomes 

S.-S>--KJ  (24) 

If  6ab  is  greater  than  90°,  tan  6^  =  Kab  is  negative,  and  Sa  —  Sb 
therefore  positive.  For  copper-iron  (Fig.  80)  Sa  —  Sb  is  thus 
always  positive. 

The  experiments  of  Le  Roux  and  of  Tait  have  shown  that  for 
lead  and  for  certain  platinum-iridium  alloys  5  is  excessively  small 
or  zero.  Hence  denoting  the  metal  lead  by  Z,  and  putting  St  =  o, 
we  have 

5.  -  S.  -  5,-  -  ld(,pa}ldt  =  -  td(tPJt)dt  -  -  t  tan  6al 

Hence,  as  a  matter  of  convenience,  lead  is  chosen  as  a  standard 
metal,  and  the  thermoelectric  lines  of  all  other  metals  and  alloys 
are,  in  general,  drawn  with  respect  to  this  metal. 

12.  The  Thermoelectric  Diagram.  In  Fig.  83  are  drawn  the 
thermoelectric  lines  with  respect  to  lead  of  a  number  of  metals 
and  alloys.  The  line  for  lead  of  course  coincides  with  the  axis  of 


THERMAL   AND    VOLTAIC    E.M.F.S. 


257 


temperatures,  and  all  other  substances  for  which  S  =  o  have  lines 
parallel  to  this  axis.  The  system  of  thermoelectric  lines  is 
known  as  the  thermoelectric  diagram. 

As  an  introduction  to  the  use  of  the  thermoelectric  diagram, 
we  shall  consider  in  detail  the  ideal  thermoelectric  lines  A  and 
B  of  two  metals  A  and  B  with  respect  to  lead,  A  being  a  straight 
line,  as  in  the  common  case,  and  B  an  irregular  curve,  Fig.  82. 
The  construction  of  the  remainder  of  the  figure  is  sufficiently 


S~  o"  Q"'  V. 

0  D  Absolute  Temperature 


Fig.  82. 

obvious,  all  the  lines  being  either  parallel  or  perpendicular  to  the 
axis  of  temperatures,  or  tangential  to  the  line  B. 

The  thermoelectric  power  of  A  with  respect  to  lead,  tpal,  is 
OAQ  at  the  absolute  temperature  zero.  From  this  value  it  regu- 
larly decreases  as  a  linear  function  of  the  temperature,  passing 
through  the  value  LA  at  the  temperature  t,  and  reaching  o  at 
the  temperature  ty  This  point  is  the  neutral  temperature  of  A 
with  respect  to  lead.  Beyond  /5,  tpal  is  negative. 

The  thermoelectric  power  of  B  with  respect  to  lead,  tpbv  is 
^  a  negative  quantity,  at  o°  absolute.  At  /0,  a  neutral  tem- 


258  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

perature  of  B  with  respect  to  lead,  tpm  =  o.  Between  /0  and  /6, 
tPbi  *s  positive,  having  a  maximum  value  at  /4.  At  t&  and  /10, 
other  neutral  temperatures  of  B  with  respect  to  lead,  tpbl  is  again 
zero,  reaching  between  them,  at  the  temperature  /8,  a  negative 
maximum. 

At  the  temperature  /,  fal  =  t  tpal  =  a  A  x  AL  —  area  ALOaA. 
When  /  =  o,  this  area  is  0  x  OAQ  =  o  ;  when  /  =  /5,  its  value  is 
OA5  x  0  =  o  ;  beyond  /5,  ALOaA  is  below  the  line  OT,  or  is 
negative.  Thus  at  tgy  tPal  =  /9  ,9/a,  =  <9/9  x  //49,  the  thermo- 
electric power  t^A^  being  negative,  and  the  area  lying  wholly 
below  OT. 

Similarly,  at  /°,  tPbl  =  t  tpbl  =  area  BLObB.  When  /  =  o,  this 
area  is  0  x  OBQ  =  o  ;  between  /  =  o  and  /  =  /0,  the  area  is  nega- 
tive, or  below  OT;  at  t^  t&,  and  tlQ,  the  neutral  temperatures  for 
B  and  lead,  the  area  is  zero  ;  from  /0  to  t&  the  area  is  positive, 
and  from  /6  to  /10  negative. 

At  *°,  tSa=  t  x  (-tan  ^  J  =AQA"  x  AA"/AQA"=AA".  Since 
tan  tOal  is  constant  and  negative  (0al  greater  than  90°  and  less 
than  1  80°),  Sa  =  AA"  is  always  positive,  or  AA"  is  always 
drawn  upward  from  A. 

The  quantity  tSa  dt  is  equal  to  AA"  x  A"  A'"  =  area  Aaa'  A'  A. 

In  like  manner,  at  t\  tSb=t  x  (-tan  fl^BJB"*  BBffjBQ'Bff 
=  BB".  When  the  point  B"  is  below  the  point  B  (t6bl  less 
than  90°),  56  =  BB"  is  negative.  Thus  from  /=o  to  /=/4, 
and  from  t  =  /7  to  /  =  /10,  56  is  negative  ;  while  from  t  =  /4  to 
/  =  /g,  56  is  positive.  At  the  temperature  /  =  //r  it  has  the  posi- 
tive value  BJ'BJ. 

The   quantity-^,  dt  is  equal   to  area  BB'B'"B"B  =  area 


At  the  temperature  /,  ^^  =  ^  -  tpbl  =LA—LB  =  BA.  BA 
is  positive,  or  A  is  above  B,  from  /  =  o  to  /  =  /3  (a  neutral  tem- 
perature for  the  thermocouple  AB)y  and  from  t  =  /7  to  /  =  /9  (ad- 
ditional neutral  temperatures  for  the  couple  ^^),  but  is  negative 
(A  below  .#)  from  /  =  /3  to  /  =  *7. 


THERMAL   AND   VOLTAIC    E.M.F.S.  259 

The  total  thermal  electromotive  force  in  a  circuit  consisting  of 
the  two  metals  A  and  B,  with  junctions  at  temperatures  t^  and 


This  result  can  be  obtained  also  from  the  relation  (13).      For 
5  _  £<fc  =  area  Aaa'  A'  A  +  area  Bbb'  B'  B 

,  (i)   +  area  MW?i  (2) 


and 

A  ~  if*  =  area  ^WA4.  (3)   ~  area  A.a^B.A,  (4) 


Hence 

A  =  (0  +  (2)  +  (3)  -  (4)  -  area  A^ 

as  proved  otherwise  above. 

If  one  junction  of  the  thermoelement  AB  is  kept  at  the  con- 
stant temperature  tv  while  the  temperature  t  of  the  other  junc- 
tion, at  first  equal  to  tv  is  gradually  increased,  ht^ab  w^  increase 
from  zero,  its  value  when  t  =  tv  until  t  =  ty  when  it  has  the  value 
Va^oft  =  area  A\Af\Ar  If  t  is  increased  beyond  /3,  to  t'  for 
instance,  the  e.m.f.  will  diminish,  since 


—  Aff^A^A^A^  (2) 

tPab  being  negative  between  /  =  /3  and  t  =  tr  When  area  (2) 
becomes  equal  in  magnitude  to  area  (i),  t^ab  is  zero,  and  t'  is 
a  temperature  of  inversion  for  A  and  B  with  respect  to  tr  As  / 
is  still  further  increased,  the  e.m.f.  increases  negatively  until  /=/7, 
beyond  which  it  increases  algebraically,  or  decreases  negatively, 
with  another  inversion  at^8'(area  BJA^A^B^'  =  ar^A^^A^A^), 
until  /  =  /9.  Beyond  this  temperature  tlfl?ab  increases  negatively, 
inverting  again  at  BIV  and  thereafter  remaining  negative. 


260          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

When  ^  =  tz  and  /=  /7,  the  Peltier  e.m.f.s  at  both  junctions 
are  zero,  no  heat  being  there  absorbed  or  evolved,  and  t^ ab  = 

A  =  (negative)  area  Af^A^  (i)  =   ^(S^-S^dt.     The 

Jts 


Temperature  in  Degrees  Centigrade 
Fig.  83. 


THERMAL   AND    VOLTAIC    E.M.F.S.  261 

e.m.f.  is  negative,  or  the  current  flows  (if  the  circuit  is  closed) 
from  B  to  A  across  the  hot  junction  at  temperature  tr 

When  t,  =  /7  and  t  =  tv  tlt^ab  =  tltTab  =  area  A^BJ'A^  (2), 
and  is  positive,  or  directed  from  A  to  B  across  the  hot  junction. 

If  tl  =  /3  and  /  =  /9,  the  Thomson  e.m.f. s  are  still  the  only 
e.m.f.s  in  the  circuit  and  t.^ab  =  (i)  —  (2).  Since  (i)  is  greater 
than  (2),  the  resultant  e.m.f.  is  negative,  or  the  current  flows  from 
B  to  A  across  the  hot  junction. 

For  an  extended  discussion  of  the  electron  theory  of  thermal 
electromotive  forces  reference  must  be  made  to  a  previously 
mentioned  article  by  Drude,  Ann.  der  Physik,  III.,  p.  369,  1900. 

13.  The  Intrinsic  E.M.F.  of  a  Reversible  Voltaic  Cell.  The 
Theory  of  Kelvin  and  von  Helmholtz.  By  a  reversible  cell  is  meant 
a  cell  in  which  all  the  processes,  both  chemical  and  physical, 
are  completely  reversed  when  the  direction  of  the  current  is 
reversed,  the  Joulean  evolution  of  heat  excepted.  Such,  for 
example,  is  a  Daniell  cell,  which  consists  of  a  zinc  electrode 
immersed  in  a  solution  of  zinc  sulphate  and  a  copper  electrode 
immersed  in  a  solution  of  copper  sulphate,  all  contained  in  the 
same  vessel,  interdiffusion  of  the  two  solutions  being  prevented 
by  a  porous  cup  between  them  or  by  other  means.  When  a 
charge  Q  (§  5,  IX.)  passes  through  the  cell  from  the  zinc  (V  =  2) 
to  the  copper  (c1 '  =  2),  one  half  gram  ion  of  zinc  goes  into  solu- 
tion and  one  half  gram  ion  of  copper  is  deposited  on  the  copper 
electrode  ;  and  when  the  same  charge  is  passed  through  the  cell 
in  the  opposite  direction,  one  half  gram  ion  of  copper  goes  into 
solution  and  one  half  gram  ion  of  zinc  is  deposited  on  the  zinc 
electrode.  The  thermoelectric  processes  occurring  at  the  con- 
tacts of  the  dissimilar  substances  are  also  reversible  with  the 
current.  The  Joulean  heat,  which  is  proportional  to  the  square 
of  the  current  and  irreversible,  may  be  made  as  small  as  desired 
in  comparison  with  the  energy  transformed  reversibly,  which  is 
proportional  to  the  first  power  of  the  current,  by  diminishing  the 
strength  of  the  current. 


262  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

Let  the  e.m.f.  of  a  reversible  cell  at  the  absolute  temperature 
/  be  denoted  by  W.  Let  the  electrodes  be  connected  up  to  the 
plates  of  a  continuously  adjustable  condenser  so  that  by  grad- 
ually diminishing  or  increasing  the  capacity  a  charge  may  be 
sent  very  slowly  in  either  direction  through  the  cell,  the  Joulean 
heat  being  made  negligible. 

Let  the  system  now  be  carried  through  a  reversible  cycle  as 
follows  :  (i)  With  the  cell  at  the  temperature  t,  let  the  capacity 
of  the  condenser  be  slowly  increased  until  a  small  charge  Q/n, 
where  n  is  a  large  number,  has  passed  through  the  cell.  The 
voltage  of  the  cell  remaining  constant  through  the  process,  ex- 
ternal work  will  be  done  by  the  cell  equal  to  WQ/n.  The 
energy  of  the  condenser  is  increased  by  ^9Qfri9  and  the  me- 
chanical energy  of  the  system  increases  by  the  same  amount 

(§  55,  !•)• 

The  source  of  the  energy  transformed  by  the  cell  is,  in  general 

partly  chemical  and  partly  thermal.  (The  law  of  Volta  for  a 
metallic  circuit  at  uniform  temperature  does  not  hold  for  a  circuit 
partly  electrolytic.)  Let  J  denote  the  net  energy  transformed 
from  chemical  into  electrical  energy  when  a  charge  Q  traverses 
the  cell  at  the  temperature  /  in  the  direction  of  the  e.m.f.  Then, 
if  /is  not  equal  to  WQ,  an  amount  of  heat 


is  absorbed  by  the  cell  during  the  above  process,  according  to 
the  principle  of  the  conservation  of  energy. 

(2)  Let  the  cell  be  cooled  to  the  temperature  t—  dt.     During 
the  process  an  amount  of  thermal  energy  which  may  be  made 
wholly   negligible   by  sufficiently  diminishing   dt,   is   abstracted 
from  the  cell. 

(3)  With  the  cell  at  the  temperature  t  +  dt,  let  the  capacity 
of  the  condenser  be  diminished  until  a  charge  Qjn  has  passed 
through  the  cell  in  the  opposite  direction.     Then,  if  H'  denotes 
the  quantity  of  heat  abstracted  from  the  cell  during  this  process, 

fft  =  Qfn  .  (\p  _  d^jdt  dt)  _  i  /»•(/_  dj\dt  dt) 


THERMAL   AND   VOLTAIC    E.M.F.S.  263 

(4)  Finally,  let  the  cell  be  heated  to  the  original  temperature 
t,  a  negligible  quantity  of  heat,  sensibly  equal  to  that  abstracted 
in  (2),  being  absorbed.  The  cycle  is  now  complete. 

Applying  the  second  law  of  thermodynamics  to  the  cycle,  we 

have 

(H-Hf}lH=dtjt 
that  is 


or 

which,  since  djjdt  is,  according   to  experiment,  sensibly   zero, 
may  be  written 

(28) 


If  d"W  jdt  is  positive,  ^Q  is  greater  than  7,  or  the  work  done  by 
the  cell  is  greater  than  the  energy  supplied  by  the  chemical  re- 
actions, and  a  quantity  of  heat  H=  WQ  —  J  is  absorbed  by  the 
cell  and  transformed  to  make  up  for  the  deficiency.  If  dW  fdt  is 
negative,  heat  is  given  out  by  the  cell.  If  dty  jdt  is  zero,  which 
is  nearly  true  in  the  case  of  a  Daniell  cell,  ^Q—J}  and  no  ther- 
mal energy  is  on  the  whole  transformed. 

(28)  may  be  written 

(29) 


which  is  von  Helmholz's  formula.  From  this  formula  the  e.m.f. 
of  a  reversible  cell  can  be  calculated  after  observing  J,  Q,  ty  and 
dty  jdt.  The  agreement  between  the  e.m.f.  calculated  in  this 
manner  and  the  e.m.f.  determined  by  direct  experiment  is,  in 
many  cases,  very  close. 

14.  Intrinsic  E.M.F.  at  a  Single  Interface.  Single  Difference 
of  Potential.  The  formula  of  von  Helmholz,  just  developed  for  a 
complete  electrolytic  cell,  which  may  contain  several  electrolytes 
and  -always  contains  two  electrodes,  with  an  intrinsic  e.m.f.  at 
each  interface  (and  sometimes  throughout  each  electrolyte)  can 


264          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

obviously  be  applied  also  to  the  intrinsic  e.m.f.  at  any  one  of  the 
interfaces.  For  example,  W  may  denote  the  intrinsic  e.m.f.  act- 
ing from  zinc  to  zinc  sulphate  in  a  Daniell  cell,  J  the  heat  de- 
veloped (if  the  energy  is  not  used  electrically)  by  the  passage  of 
one-half  gram  ion  of  zinc  from  the  solid  state  into  zinc  ions,  and 
d"ty  jdt  the  temperature  rate  of  change  of  the  e.m.f.  Then 


The  actual  e.m.f.  of  the  complete  cell  is  equal  to  the  algebraic 
sum  of  all  such  included  e.m.f.s,  and  the  actual  temperature  co- 
efficient to  the  algebraic  sum  of  all  the  individual  temperature 
coefficients. 

The  intrinsic  e.m.f.  acting  from  the  zinc  (or  other  substance)  to 
the  electrolyte  will  develop  a  potential  difference  equal  to  M*  in 
magnitude  but  directed  from  the  electrolyte  to  the  zinc  (or  other 
substance).  Such  a  potential  difference  is  called  a  single  differ- 
ence of  potential.  There  is  no  satisfactory  method  of  determining 
experimentally  such  a  difference  of  potential,  nor  can  it  be  com- 
puted from  the  above  equation,  since  J  can  not  be  determined  for 
any  one  kind  of  ions,  as  zinc  ions,  alone.  For  when  one  kind  of 
ions  goes  into  solution,  another  kind  goes  out  of  solution. 

(29)  is  seen  to  include  (19)  as  a  particular  case. 

For  additional  information  on  the  theory  of  the  voltaic  cell, 
single  potential  differences,  etc.,  reference  must  be  made  to  treat- 
ises on  electrochemistry,  where  also  references  to  the  original 
literature  may  be  found. 


CHAPTER    XI. 

MAGNETS.     MAGNETOSTATIC  FIELDS. 

1.  Magnets.     A  bar  of  steel  placed  in  a  long  helix  of  wire  tra- 
versed by  an  electric  current  is  found,  on  removal  from  the  helix, 
to  have  acquired  certain  properties  analogous,  in  many  respects, 
to  those  of  an  electret,  and  is  said  to  have  been  magnetised  or  to 
have  become  a  magnet.     The  same  name  is  applied  to  any  body 
possessing  these  properties,   however  acquired,   some  of  which 
will  be  described  in  the  following  pages. 

2.  Electric  and  Magnetic  Analogues.     Just  as  an  electret  and 
the  region  outside  it  are  the  seat  of  electric  induction  or  dis- 

o 

placement,  so  a  magnet  and  the  region  around  it  possess  mag- 
netic induction.  To  the  electrisation  of  the  one  corresponds  the 
magnetisation  of  the  other.  Tubes  of  induction  run  through  the 
magnet,  entering  at  one  end  and  issuing  at  the  other,  being 
continuous  like  an  electret' s  tubes  of  displacement.  There  is, 
however,  no  magnetic  analogue  of  a  true  electric  charge,  or  dis- 
continuity of  electric  displacement.  Magnetic  conduction  and 
magnetic  conductors  do  not  exist,  a  tube  of  induction  cannot  be 
cut  in  two  thus  developing  true  magnetic  charges,  hence  the  in- 
duction is  always  continuous  or  the  tubes  of  induction  are  always 
closed.  Analogous  to  the  electric  intensity  is  the  magnetic  in- 
tensity, connected  with  the  induction  by  a  relation  similar  to  that 
which  connects  electric  intensity  and  displacement.  Substances 
differ  magnetically,  or  possess  different  inductivities,  just  as  they 
possess  different  permittivities.  Discontinuities  in  the  magnetic 
intensity  occur  where  the  tubes  of  induction  pass  from  one  sub- 
stance to  another  of  different  inductivity  or  through  a  substance 
whose  inductivity  continuously  changes.  Where  these  discon- 

265 


266  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

tinuities  of  intensity  exist  are  apparent  magnetic  charges,  or  quan- 
tities of  magnetism,  analogous  to  apparent  electric  charges,  arising 
from  discontinuities  in  the  electric  intensity.  The  regions  or 
surfaces  in  which  the  discontinuities  occur  are  called  the  poles  of 
the  magnet.  If  the  magnet  is  long  and  slender  and  the  induction 
within  it  uniform  and  in  the  direction  of  its  length,  the  poles  are 
approximately  concentrated  at  its  ends.  In  every  case,  however, 
poles  are  more  or  less  distributed. 

Upon  the  poles  of  a  magnet  in  a  magnetic  field,  as  upon  the 
poles  of  an  electret  in  an  electric  field,  forces  are  found  to  exist, 
and  from  these  forces  and  their  seats  the  strengths  of  the  poles 
are  defined  and  their  distribution  determined,  greater  or  less  forces 
corresponding,  in  general,  to  greater  or  less  pole  strengths,  and 
more  or  less  restricted  seats  of  the  forces  to  more  or  less  con- 
centrated poles. 

3.  Positive  and  Negative  Magnetic  Poles.    That  pole  of  a  mag- 
net toward  which,  while  in  the  magnetising  helix,  a  right-handed 
screw  placed  with  its  axis  coincident  with  that  of  the  helix  would 
have  to  be  translated  in  order  to  rotate  in  the  direction  of  the 
current  around  the  helix  is  called  the  positive  pole  of  the  magnet, 
and  the  other  pole  the  negative  pole.     The  terms  positive  and 
negative  in  this  connection  are  purely  conventional,  but  are  justi- 
fied as  in  electrostatics  (§  I,  I.). 

4.  The  Earth's  Magnetic  Meridian  at  a  Point.     The  Axis  of  a 
Magnet.     North  and  South  Magnetic  Poles.     A  magnet  so  sus- 
pended or  otherwise  supported  near  the  earth  as  to  have  perfect 
freedom  of  motion  about  its  center  of  gravity  will  always  take 
up  a  position  with  a  definite  line  connecting   the   positive  and 
negative  poles,  or  rather  a  definite  direction  in  the  magnet,  point- 
ing in  a  definite  geographical   direction.     This   direction  is,  in 
general,  northerly  and  southerly,  and  the  vertical  plane  through  it 
is  called  the  earth 's  magnetic  meridian  at  the  point.     The  posi- 
tive and  negative  poles  point  in  the  northerly  and  the  southerly 
direction,  respectively,  and  are  therefore  called  north  and  south 


MAGNETS.     MAGNETOSTATIC   FIELDS.  267 

poles,  respectively.  The  definite  direction  in  the  magnet,  from 
the  negative  to  the  positive  pole,  is  called  the  a&is  of  the  magnet. 
This  term  is  also  applied  to  a  certain  line,  with  this  direction,  in 
the  magnet  (§  24). 

5.  The  Force  Between  Two   Magnetic   Poles.      Between   two 
magnetic  poles  a  force  always  exists,  repulsive  or  positive  if  the 
two  poles  have  the  same  sign,  and  attractive  or  negative  if  the 
poles  have  opposite  signs. 

This  force  between  two  magnetic  poles  can  be  measured  by 
any  kind  of  a  dynamometer  provided  that  the  magnets  are  so 
long  and  their  poles  so  nearly  concentrated  at  the  ends  that  the 
poles  under  experiment  are  sensibly  outside  the  field  of  influence 
of  the  other  poles. 

The  force  decreases  rapidly  with  the  increase  of  the  distance 
between  the  poles. 

At  a  given  distance  apart  the  force  between  the  poles  depends 
upon  the  medium  in  which  they  are  immersed. 

At  a  given  distance  apart  and  in  a  given  medium  the  force  is 
different  for  different  pairs  of  poles. 

6.  Coulomb's  Law  of  Force  for  Concentrated  Poles  of  Permanent 
Magnets  in  an  Infinite  Homogeneous  Isotropic  Medium,     Consider 
three  approximately  concentrated  poles  A,  B,  C,  belonging  to 
three  very  long,  very  slender,  cylindrical  magnets  of  very  hard 
steel  magnetised  in  a  slender  solenoid  or  helix,  longer  than  the 
magnets,  traversed  by  an  electric  current.     The  more  nearly  the 
poles  are  concentrated,  the  greater  the  ratio  of  the  length  to 
the  diameter  of  each  magnet,  the  greater  the  coercive  intensity 
of  the  steel  (§   25,   XIII.),  and  the  more  nearly  homogeneous 
and  isotropic  the  surrounding  medium,  the  more  nearly  are  the 
following  relations  found  by  experiment  to  be  true  : 

The  force  between  any  two  of  the  poles  varies  inversely  as  the 
square  of  the  distance  between  them. 

The  force  between  two  of  the  poles,  as  A  and  Bt  at  any  dis- 
tance d  apart  in  a  given  medium  I  bears  to  the  force  between  the 


268          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

same  poles  at  the  same  distance  d  apart  in  another  medium,  2, 
the  same  ratio  (p2/v<},  see  below)  which  the  force  between  another 
pair,  as  A  and  C,  at  any  given  distance  d'  apart  in  medium  I 
bears  to  the  force  between  A  and  C  at  the  same  distance  d'  apart 
in  medium  2. 

If  FM  denotes  the  force  between  the  poles  A  and  C  in  any 
medium  at  any  distance  d  apart,  and  Fbc  the  force  between  the 
poles  B  and  C  in  the  same  medium  at  the  same  distance  d  apart, 
then,  whatever  this  distance  d,  and  whatever  the  medium,  Fac 
bears  to  Fbr  a  certain  definite  and  constant  ratio,  and  this  ratio  is 
unaltered  if  the  pole  C  is  exchanged  for  any  other  similar  (con- 
centrated) pole. 

Hence  there  is  associated  with  each  of  the  two  poles  A  and  B 
a  constant,  which  we  shall  call  its  pole  strength,  or  quantity  of 
magnetism,  such  that  the  force  between  one  of  these  poles  and 
a  third  concentrated  pole  is  proportional  to  this  constant  or  pole 
strength.  But  the  same  thing  is  true  of  the  third  pole  also. 
Hence  the  force  between  two  poles  is  proportional  to  the  strength 
of  each,  that  is  to  the  product  of  their  pole  strengths.  The 
strength  of  a  pole  will  de  denoted  by  m  with  a  subscript  to  iden- 
tify the  pole  when  necessary. 

Putting  the  above  results  together,  we  have,  if  F  denotes  the 
force  between  two  concentrated  poles  of  strengths  m^  and  mz 
when  the  distance  between  them  is  L,  the  law  of  Coulomb  : 

F=Amlm2/pL2  (i) 

H  being  a  constant  of  the  medium  in  which  the  poles  arc  im- 
mersed, called  its  inductivity  (analogous  to  electric  permittivity), 
and  A  a  constant  depending  on  the  units  in  which  all  the  other 
quantities  are  expressed. 

If,  as  in  this  book,  the  c.g.s.  system  of  mechanical  units  is 
adopted,  if  n  is  expressed  in  terms  of  the  inductivity  of  free 
aether  (denoted  by  /XQ)  as  unit  inductivity,  and  if  A  is  put  equal  to 
-f  i  /47T,  ml  and  mv  are,  by  definition,  expressed  in  terms  of  the 


MAGNETS.     MAGNETOSTATIC    FIELDS.  269 

rational  electromagnetic  (REM)  unit  magnetic  pole  strength.     With 
these  conventions,  the  above  equation  (i)  becomes 

F=  mlm2/47riJLL2  (2) 

If  mv  and  m2  are  poles  of  the  same  kind  (both  positive  or  both 
negative),  F  is  positive  or  repulsive  ;  if  the  signs  of  the  poles  are 
opposite,  F  is  negative,  or  the  force  is  one  of  attraction. 

From  the  nature  of  the  conditions  assumed  above  it  is  clear 
that  the  law  can  not  be  established  rigorously  by  direct  experi- 
ment. The  best  proof  of  the  law  is  the  general  agreement  be- 
tween experiment  and  a  magnetic  theory  based  largely  upon  the 
law.  The  physical  reason  for  the  existence  of  the  law  of  inverse 
squares  is  similar  to  that  for  the  law  of  inverse  squares  in  electro- 
statics, as  will  be  apparent  when  the  magnetic  flux  and  Gauss's 
theorem  for  magnetism  have  been  discussed  (§§  13  and  14). 

7.  The  Magnetic  Field.  Magnetic  Intensity.  Any  region  in 
which  a  magnetic  pole  is  acted  upon  by  a  mechanical  force  in 
virtue  of  its  magnetism  is  called  a  magnetic  field.  Such  a  field  is 
the  neighborhood  of  a  magnetic  pole,  or  the  region  about  the 
earth,  or  the  region  about  a  wire  carrying  an  electric  current. 

It  follows  from  experiment  that  the  force  F  acting  upon  a  con- 
centrated magnetic  pole  at  any  point  of  a  magnetic  field  is  pro- 
portional to  its  pole  strength  m  (provided  that  the  distribution  of 
magnetism  originally  accompanying  the  field  remains  sensibly 
unaltered  on  the  introduction  of  the  pole).  That  is, 

F-Hm  (3) 

The  proportionality  factor  H  is  called  the  magnetic  intensity  at 
the  point.  H  is  evidently  a  vector  specifying  the  state  of  the 
field  at  the  point,  its  direction  being  that  of  the  force  upon  a 
positive  pole  (or  opposite  to  that  of  the  force  upon  a  negative 
pole),  and  its  magnitude  the  magnitude  of  the  force  upon  unit 
pole  placed  at  the  point. 

When  F  is  expressed  in  dynes  and  m  in  the  REM  unit,  H  = 
F/m  is,  by  definition,  expressed  in  the  REM  unit  magnetic  in- 
tensity. 


2/0          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

(2)  is  a  particular  case  of  (3). 

The  term  magnetic  field  is  also  used  to  denote  the  collective 
intensity  in  a  region  instead  of  the  region  itself.  The  direction 
of  the  field  at  any  point  is  then  the  direction  of  the  intensity,  and 
the  strength  of  the  field  \s,  the  magnitude  of  the  intensity. 

8.  The  Strength  of  a  Distributed  Pole,     By  (3),  which  may  be 
written 

m  =  FjH  ;_.     (4) 

the  strength  of  any  magnetic  pole  when  placed  in  a  uniform  field 
may  be  defined  as  the  ratio  of  the  force  acting  upon  the  pole  to 
the  magnetic  intensity  of  the  field  (which  can  be  determined  by 
(3)  with  a  concentrated  pole).  A  general  definition  of  pole 
strength  consistent  with  (2)  and  (4)  is  given  in  §  21. 

9.  The  Positive  and  Negative  Poles  of  a  Magnet  Have  the  Same 
Numerical  Strength,  or  the  total  quantity  of  magnetism  in  any 
magnet  is  zero.     For  if  a  magnet  is  placed  on  a  float  in  a  vessel 
of  water,  so  as  to  be  perfectly  free  to  move  in  any  direction,  it 
experiences  no  translatory  force  in  any  direction  when  the  sur- 
rounding field  is  that  of  the  earth  alone.     Thus  the  force  upon 
the;  positive  pole  is  exactly  equal  and  opposite  to  the  force  upon 
the  negative  pole.      Hence,  since  the  intensity  of  the  earth's  field 
throughout  the  region  occupied  by  the  magnet  (and,  in  general, 
throughout  a  much  larger  region)  is  sensibly  uniform,  the  pole 
strengths  are  equal  and  opposite  by  (4),  and  their  algebraic  sum 
is  zero.     See  also  §  21. 

10.  Magnetic  Induction.     Analogous  to  electric  induction  or 
displacement  is  a  quantity  called  the  magnetic  induction,  defined 
as  the  product  of  the   inductivity  and  the  magnetic  intensity. 
Thus,  if  the  magnetic  induction  is  denoted  by  B,  we  have 

B  =  »H  (5) 

B  is  obviously  a  vector  with  the  same  direction  as  that  of  H, 
since  p  (in  isotropic  media,  which  alone  will  be  considered  here) 
has  no  relation  to  direction. 


MAGNETS.     MAGNETOSTATIC    FIELDS. 

When  /JL  and  H  are  expressed  in  REM  units,  B  —  /z  H  is  said 
to  be  expressed  in  the  REM  unit  magnetic  induction. 

A  substance  in  which  magnetic  induction  exists,  that  is,  a  sub- 
stance which  supports  a  magnetic  field,  is  said  to  be  magnetised  or 
to  be  in  a  state  of  magnetisation.  If  the  induction  and  inductivity 
are  uniform  throughout,  the  magnetisation  is  said  to  be  uniform. 
The  intensity  of  magnetisation  is  defined  in  §21. 

11.  Lines  and  Tubes  of  Magnetic  Intensity  and  Induction  are 

defined  in  exactly  the  same  way  as  lines  and  tubes  of  electric 
intensity  and  induction,  except  that  magnetic  quantities  are  sub- 
stituted for  electric  throughout. 

12.  The  Superposition  of  Magnetic  Fields.     The  statements  of 
§12,  I.,  and  those  of  the  paragraph  following  (2),  §  n,  I.,  with 
reference  to  the  electric  field  hold  also  for  the  magnetic  fielcl,  ex- 
cept that  the  medium  supporting  a  magnetic  field  never  breaks 
down  as  a  result  of  magnetic  stress  (another  illustration  of  the 
non-existence  of  magnetic  conductivity). 

13.  The  Magnetic  Flux,  <3>,  across  a  surface  5  is  defined  in  ex- 
actly the  same  manner,  analogous  terms  being  substituted,  as 
the  electric  flux,  §  22,  I.,  that  is,  as  the  integral  over  the  surface 
of  the  normal  component  of  the  induction.     Thus 

d>=  §B  cos  e  dS  =/  fiffcos  6  dS  (6) 

0  denoting  the  angle  between  B  or  H  and  the  normal  to  dS. 

14.  Gauss's  Theorem.     The  magnetic  flux  <f>  outward  across  a 
closed  surface  5"  surrounding  any  number  of  concentrated  mag- 
netic poles  of  total  strength  m  is  equal  to  m.*     This  follows  for 
an  infinite  or  finite  region  containing  a  homogeneous  isotropic 
medium  (/u  =  constant,  except  within  the   magnets,  whose  vol- 

*  With  the  qualification  made  below.  In  every  case,  however,  the  flux  of  mag- 
nectic  intensity  outward  across  a  closed  surface  surrounding  a  real  pole  distributed  in 
any  manner,  multiplied  by  the  inductivity  of  the  medium  surrounding  the  magnet, 
is  equal  to  the  strength  of  the  pole.  For  the  analogous  electric  case  see  VI.  and  IV. 


2/2  ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

umes  are  supposed  negligible)  from  (2),  (3),  (5),  and  (6),  and  a 
process  of  reasoning  exactly  analogous  to  that  employed  in  es- 
tablishing the  corresponding  theorem  in  electrostatics. 

The  theorem  holds  only  for  concentrated  poles,  the  flux  from 
a  distributed  pole  being,  in  general,  very  different  from  m.  (See 
§  8,  VI.,  for  the  corresponding  electric  case.) 

The  strength  of  a  tube  of  induction.  It  follows  from  Gauss's 
theorem  exactly  as  in  electrostatics  that  the  flux  across  every 
diaphragm  of  a  given  tube  of  magnetic  induction  in  a  homogene- 
ous isotropic  medium  is  the  same.  The  magnitude  of  this  flux 
is  called  the  strength  of  the  tube.  A  unit  tube  is  a  tube  whose 
strength  is  unity. 

We  here  assume,  as  in  strict  accord  with  experiment,  that  the 
strength  of  a  tube  is  constant  throughout  its  length  whatever 
media  it  may  traverse,  whether  the  field  is  a  pure  magnetic  field 
or  'an  electromagnetic  field  (XII.).  For  a  pure  magnetic  field 
(in  which  there  is  no  intrinsic  magnetisation)  this  result  follows 
from  considerations  exactly  similar  to  those  adduced  to  establish 
the  analogous  proposition  in  electrostatics,  except  that  two  con- 
centrated permanent  magnetic  poles,  one  in  each  medium,  must 
be  employed  instead  of  the  closed  electric  condenser. 

In  deriving  Gauss's  theorem  the  (infinitely  small)  volumes  of 
the  magnets  and  all  their  contents  were  neglected.  It  must 
always  be  remembered,  however,  that,  as  stated  in  §  2,  magnetic 
poles  are  the  analogues  of  the  poles  of  electrets,  not  of  true  elec- 
tric charges.  We  may,  for  convenience,  consider  only  the  flux 
from  (or  to)  a  pole  in  the  surrounding  medium,  as  we  have  just 
done ;  but  we  must  remember  that  the  same  quantity  of  flux 
which  emanates  from  a  magnet  at  its  positive  pole  enters  the 
magnet  again  at  its  negative  pole,  making  the  total  flux  across 
any  closed  surface  surrounding  a  single  pole  (or  any  number  of 
poles)  zero. 

That  this  statement  is  correct  for  a  magnet  with  concentrated 
poles  follows  from  Gauss's  theorem  and  the  fact  that  if  any  mag- 
net is  broken  across  its  axis  into  any  number  of  pieces,  each 


MAGNETS.     MAGNETOSTATIC    FIELDS.  2/3 

piece  is  a  magnet  with  its  positive  and  negative  poles  equal  in 
strength  (numerically)  and  pointing  in  the  same  directions  as  the 
corresponding  poles  of  the  original  magnet. 

When  both  the  original  magnet  and  these  pieces  are  very  long 
and  thin,  as  they  must  be  to  have  approximately  concentrated 
poles,  the  pole  strengths  of  the  pieces  and  the  original  magnet 
are  sensibly  equal. 

We  here  assume  that  all  tubes  of  magnetic  induction  are  closed 
like  the  tubes  just  considered,  whether  in  a  pure  magnetic  field 
or  in  an  electromagnetic  field.  This  assumption  is  in  strict 
accord  with  experiment  and  with  a  more  general  definition  of 
magnetic  induction  given  in  Chapter  XIII.  Thus  there  is  noth- 
ing in  magnetism  analogous  to  the  discontinuity  of  electric  dis- 
placement, or  true  electric  charge. 

Applying  the  above  results  to  the  element  of  volume  at  a 
point,  we  get,  obviously, 


div  B  =  o  =  div  pH  (7) 

the  flux  into  any  element  of  volume  across  a  part  of  its  surface 
being  equal  to  the  flux  out  of  the  volume  across  the  rest  of  the 
surface. 

15.  Magnetomotive  Force  or  Gaussage.  Magnetic  Potential 
and  Equipotential  Surfaces.  The  line  integral  of  magnetic  inten- 
sity, ^  H  cos  6  dLy  along  a  path  L  from  a  point  Pl  of  a  magnetic 
field  to  a  point  P2  is  called  the  magnetomotive  force  (m.m.f.)  or 
gaussage  from  Pl  to  P2  along  the  path  L.  When  this  integral  is 
the  same  along  every  path  from  Pl  to  P2  it  is  also  called  the 
difference  of  magnetic  potential  between  Pl  and  P2,  or  the  fall  of 
magnetic  potential  from  Pl  to  P2.  From  a  process  of  reasoning 
similar  to  that  of  §  17,  L,  this  is  evidently  the  case  in  the  field 
of  a  magnet  (unaccompanied  by  electric  currents). 

The  unit  gaussage  is  the  gaussage  which  exists  between  two 
points  when  unit  work  must  be  done  to  transfer  a  unit  magnetic 
pole  from  one  to  the  other. 


2/4          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

The  fall  of  magnetic  potential  from  a  given  point  P  to  a  point 
at  an  infinite  distance  from  all  magnetic  poles  is  called  the  mag- 
netic potential  at  P,  and  will  be  denoted  by  H.  This  symbol  will 
also  be  used  to  denote  m.m.f.  in  the  more  general  case. 

A  surface  which  is  everywhere  normal  to  the  magnetic  intensity 
is  called  a  magnetic  equipotential  surface. 

16.  The  Mapping  of  Magnetic  Fields.    A  magnetic  field  can  be 
completely  mapped  out  by  a  system  of  tubes  of  induction  of 
equal  strength  or  by  a  system  of  equipotential  surfaces  between 
the  successive  surfaces  of  which  the  gaussage  is  constant.     (See 
§§20  and  25,  I.) 

Maxwell's  method  of  drawing  such  systems  of  tubes  and  sur- 
faces applies  equally  to  the  electric  and  magnetic  cases.  (See 
§§  7,  n,  13  and  14,  II.) 

17.  Magnetic  Conductors.     An  imaginary  substance,  analogous 
otherwise  to  an  electric  conductor,  within  whose  volume  there  is 
no  magnetic  induction  or  intensity  when  immersed  in  a  static 
magnetic  field,  and  at  whose  surface  all  lines  of  magnetic  inten- 
sity and  induction  are  therefore  discontinuous  normally,  is  called 
a  magnetic  conductor.     It  will  appear  later  (XVI.)  that  a  perfect 
electric  conductor  behaves  like  a  substance  of  zero  inductivity 
and  can  under  no  circumstances  support  an  electric  or  magnetic 
field. 

The  (imaginary)  true  magnetic  charge  and  surface  density  upon 
such  an  imaginary  surface  are  defined  as  the  magnetic  flux  from 
surface  and  the  induction  at  the  surface,  respectively,  in  a  manner 
exactly  analogous  to  that  followed  in  electrostatics. 

18.  Magnetic  Energy  Density,  Magnetic  Tension,  and  Magnetic 
Pressure.     From  the  strict  mathematical  analogy  existing  between 
the  strength  of  a  concentrated  permanent  magnetic  pole  and  a 
concentrated  electric  charge  or  the  concentrated  pole  of  a  per- 
manent electret,  magnetic  inductivity  and  electric  permittivity, 
magnetic  intensity  and  electric  intensity,  magnetic  induction  and 


MAGNETS.     MAGNETOSTATIC    FIELDS.  275 

electric  induction  or  displacement,  it  follows  that,  when  ft  is  con- 
stant, 

(i)  The  magnetic  energy  per  unit  volume  at  any  point  of  a 
magnetic  field  is 


(2)  There  is  a  tension  parallel  to  the  intensity  equal  to 

r=  \BH=^  pH2  =  1  B*jn  (9) 

(3)  There  is  a  pressure  in  every  direction  normal  to  the  inten- 

sity equal  to  ..  „  N 

T=\BH  =  \  nH~  =  etc.  (10) 


It  follows  also,  whether  //,  is  constant  or  not,  that  the  work 
per  unit  volume  done  in  magnetising  a  substance  from  a  value 
of  the  induction  B  =  Bl  to  a  value  B  =  B2  is 


dWjdT=\     HdB  (n) 

which  reduces  to  (8)  when  /*  is  independent  of  H.  This  expres- 
sion does  not  give  the  change  in  the  magnetic  energy  when  hys- 
teresis (§  39,  XIII.)  is  present,  by  far  the  greater  part  of  the 
energy  used  during  the  magnetising  process  being  in  most  cases 
dissipated. 

(8)  and  (n)  will  be  demonstrated  later  (§§  12  and  29,  XIII.) 
in  a  different  manner. 

19.  Magnetic  Energy.  Permeance.  Reluctance.  In  exactly 
the  same  manner,  or  by  direct  integration  from  (8),  the  energy  con- 
tained in  the  volume  T  of  a  tube  of  magnetic  induction  of  strength 
<I>  between  two  equipotential  surfaces  between  which  the  m.m.f. 


if  p  is  constant  (independent  of  H) ;  and 

fffHdLdSdB 

(14) 
SdB  = 


2/6    ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

in  the  general  case,  assuming  no  dissipation  of  energy.     (14) 
gives  the  work  done  in  magnetisation  in  every  case. 

The  ratio  of  the  magnetic  flux  <l>  through  the  tube  to  the 
m.m.f.  fl  between  the  two  equipotentials  is  called  the  permeance, 
and  its  reciprocal  the  reluctance,  of  this  portion  of  the  tube. 
Thus,  if  the  permeance  is  denoted  by  Pand  the  reluctance  by  R, 

P=i/R=3>/£l  (15) 

The  combination  of  (13)  and  (15)  gives  for  the  energy  IV, 

W=  ifl<S>  =  IPS?  =  13>'2JP  =  I&IR  =  i^4>2         (16) 

The  electric  analogue  of  permeance  is  evidently  permittance. 
[In  the  irrational  systems  of  units  also,  Chapter  XIV.,  Pand 
R  are  defined  by  (15),  the  electrical  analogy,  requiring  the  intro- 
duction of  477-,  not  being  strictly  maintained.] 

20.  The  Laws  of  Refraction  of  Lines  of  Intensity  and  Induction. 
It  follows  also,  by  a  procedure  exactly  analogous  to  that  of  IV., 
§  2,  or  by  inspection  of  the  final  equations  of  IV.,  §  2,  and  the 
analogies  mentioned  in  §  2,  that  in  crossing  an  interface  from  a 
medium  I  to  a  medium  2  a  line  of  magnetic  intensity  or  induction 
is  refracted  in  such  a  way  that 

I.  The  incident  and  refracted  lines  are  in  the  same  plane  per- 
pendicular to  the  interface  at  the  point  of  incidence ;  and  that 

II.  The  ratio  of  the  tangent  of  the  angle  of  incidence  to  the 
tangent  of  the  angle  of  refraction   is  a  constant  for  the  given 
media  (when  the  inductivity  ratio  is  constant)  and  equal  to  the 
ratio  of  the  two  inductivities. 

The  equivalent  equations  are 

HI  sin  0L  =  H2  sin  02  (tangential  intensity  continuous)  (17) 
Bl  cos  6l  =  B2  cos  *02  (normal  induction  continuous)  (18) 
tan  0j/tan  02  = /A^  (19) 

21.  Magnetic  Surface   and  Volume  Density,  Intensity  of  Mag- 
netisation, etc.     Proceeding  in  the  same  manner  and  following 
IV.,  §§  3-4,  we  get  what  follows. 


MAGNETS.     MAGNETOSTATIC    FIELDS.  277 

The  normal  discontinuity  in  the  magnetic  intensity  at  the  inter- 
face separating  two  media  I  and  2  (H^  and  H2  being  reckoned 
positive  when  directed  from  medium  2  to  medium  i)  is 

HI  cos  0l  -  H2  cos  ez 

which  is  equal  to  [(/*2  —  /-OM/^]^  cos  ^2  wnen  there  is  no  in- 
trinsic magnetisation  (§  22). 

The  magnetic  surface  density  (analogous  to  apparent  electric 
surface  density)  at  the  interface  is 

*>  =  ^(ff,  cos  0l  -  H2  cos  02)  =  (B2  -  ^H2}  cos  62     (20) 

which  is  equal  to  [_(^z  —  /O/A^]^  cos  ^2  wnen  there  is  no  intrin- 
sic magnetisation  present.  [In  the  irrational  systems  of  units, 
Chapter  XIV.,  <rf  is  defined  by  the  relation 

47TCT'  =  /^(//j   COS   0l  -  H2  COS   ft,)] 

The  intensity  of  magnetisation  of  medium  2  with  respect  to 
medium  I  is,  by  definition, 

J=B2-^H2  (21) 

which  is  equal  to  [Gv/'O/pJ-^j  when  there  is  no  intrinsic  mag- 
netisation. An  equivalent  definition  of  J  is  the  magnetic  moment 
of  medium  2  per  unit  volume  (§23,  and  §  12,  IV.).  (21)  shows 
that  J  is  the  difference  between  the  actual  induction  in  2  and  the 
induction  which  would  exist  there  if  /i2  were  equal  to  /^  with  the 
same  value  of  the  intensity.  [In  the  irrational  systems  of  units 
(Chapter  XIV.),  /is  defined  by  the  equation  47r/=  •B2—fAlffy'] 
If  we  write  B2  =  p2Hz,  (21)  may  be  written 


(22) 

*,  which  is  written  for  (^—fj,^,  is  called  the  magnetic  suscepti- 
bility of  medium  2  with  respect  to  medium  I  .     [In  the  irrational 
system  of  units  K  =  (/*2  —  <"1)/47r.] 
(22)  may  be  written 

(23) 


2/8          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

(20)  may  be  written 

a'=J  cos  02  (24) 

The   quantity  of  magnetism   upon   a   surface   S,   or  the  pole 
strength  of  the  surface  S,  is 

JS=fjcos0jiS  (25) 

which  may  be  written  equal  to  \_(^2  —  A^)//^]  3>  when  there  is  no 
intrinsic  intensity. 

The  magnetic  volume  density  in  a  region  where  /-t  varies  from 
point  to  point  is  p>  =  ^  div  H       .&<&&*&$     (26) 

(In    the  irrational   systems  of  units,   Chapter  XIV.,  47rp'  = 
A*!  div  H.) 

The  total  quantity  of  magnetism  in  a  volume  T  is 

fp'dr  =  ^f  div  //  dr  (27) 

The  total  quantity  of  magnetism  within  a  volume  r  and  upon 
its  surface  5  is 

m  =f<r'dS  +fp'dr  =  //  cos  02  dS  -f  ^  /div  J7  rfr     (28) 

The  force  upon  a  magnetic  pole  of  strength  m  in  a  field  of 
intensity  //is  zr 


/     x 
(29) 

The  intensity  at  a  point  /*  distant  L  from  the  element  <^w  of  a 
pole  of  strength  dm  "due"  to  dm  is  directed  along  L  and  is  equal  to 

dH  =  dml^TT^L2  (30) 

The  magnetic  potential  at  a  point  due  to  any  magnetic  distri- 
bution whatever  is 

II  =  I/4W/4J  -fdm/L  (31) 

If  medium    2    is    uniformly    magnetised,   the    magnetic    flux 
through  the  positive  pole  will  be 

cos  02dS  =fj  cos  02dS  +  ^  fH2  cos  BjtS 

(32) 


MAGNETS.     MAGNETOSTATIC    FIELDS.  279 

which  is  equal  to  m  only  when  the  pole  is  concentrated.  In 
isolated  magnets  H2  is  negative,  hence  <S>  is  less  than  m  (except 
in  the  ideal  case  mentioned). 

The  complete  developments  §  7,  IV.,  and  §  I,  V.,  are  valid  for 
the  magnetic  case,  magnetic  quantities  being  substituted  for  the 
analogous  electric  quantities. 

22.  Intrinsic  Magnetisation,  Etc.  Corresponding  to  intrinsic 
electrisation,  intrinsic  electric  intensity,  etc.,  are  intrinsic  magnet- 
isation, of  which  we  have  an  example  in  every  magnet,  intrinsic 
magnetic  intensity  or  force  (denoted  by  Ji)  maintaining  the  induc- 
tion, etc.  Intrinsic  magnetic  phenomena  do  not  appear  in  most 
substances,  but  are  far  more  marked  than  the  corresponding  elec- 
tric phenomena  in  others,  notably  iron,  nickel,  and  cobalt,  reach- 
ing their  maximum  development  in  hard  steel.  The  magnetisation 
of  these  substances  will  be  briefly  discussed  in  §§  35-39,  XIII. 

Permanent  Magnets.  A  permanent  magnet  is  a  magnet  whose 
pole  strengths  at  any  definite  temperature  are  constant  independ- 
ently of  the  time,  of  the  nature  of  the  surrounding  medium,  and 
of  the  proximity  of  other  magnets  and  electric  currents.  Such  a 
magnet  is  purely  ideal,  but  approximately  permanent  magnets 
can  be  made  of  long  cylinders  of  properly  tempered  hard  steel. 
Such  a  magnet,  if  not  brought  into  too  strong  magnetic  fields, 
and  if  kept  when  not  in  use  in  an  iron  case  will  retain  its  moment 
constant  in  air,  even  when  the  ratio  of  its  length  to  the  square  root 
of  its  cross -section  is  less  than  ten  to  one,  for  at  least  a  year  within 
one  tenth  per  cent.  (Klemencic,  Ann.  der  Physik,  XII. ,  174, 
1901).  From  the  discussion  of  the  permanent  electret,  §  8,  VI., 
it  is  clear  that  a  magnet  will  be  more  nearly  permanent  the 
greater  the  ratio  of  its  length  to  its  diameter  (or  other  linear  di- 
mensions perpendicular  to  its  length).  It  is  also  evident  that, 
other  things  being  equal,  the  greater  the  coercive  force  (§25, 
XIII.)  of  a  substance  or  the  "harder"  it  is  magnetically,  the 
nearer  will  be  its  approach  to  permanency  of  magnetisation, 
when  made  up  into  magnets. 


280          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

23.  Pure  Magnetic  Fields.  It  is  now  evident  that  the  magnetic 
fields  for  a  great  variety  of  ideal  distributions  can  be  obtained 
from  the  analogous  electric  fields  already  discussed  in  previous 
chapters,  by  substituting  magnetic  conductors  for  electric  con- 
ductors, inductivity  for  permittivity,  true  magnetic  charge  for  true 
electric  charge,  magnetic  pole  strength  for  fictitious  electric 
charge,  magnetic  intensity  for  electric  intensity,  magnetic  induc- 
tion for  electric  displacement,  intensity  of  magnetisation  for  in- 
tensity of  electrisation,  etc.  Since  the  concentrated  poles  of 
permanent  electrets  can  be  substituted  for  concentrated  electric 
charges,  the  electrisation  within  the  volume  of  the  electrets  being 
neglected  (§  8,  VI.),  concentrated  electric  charges  can  be  re- 
placed in  the  magnetic  analogues  by  concentrated  magnetic  poles, 
the  magnetic  induction  within  the  volumes  of  the  linear  magnets 
being  neglected.  The  approximate  effects  of  magnetic  conduc- 
tors, except  as  regards  the  continuity  or  discontinuity  of  the 
magnetic  induction,  can  be  obtained  by  using  substances  of  great 
inductivity.  (The  value  of  p  for  soft  iron  may,  with  vibration, 
reach  nearly  80,000  /u,().  See  Ewing,  Phil.  Trans.,  1885.) 

Thus,  Fig.  14  represents  the  field  of  a  single  concentrated 
magnetic  pole,  the  induction  within  the  magnet,  supposed  infin- 
itely thin,  being  neglected,  and  the  opposite  pole  being  supposed 
infinitely  remote. 

Fig.  22  shows  the  field  surrounding  a  single  magnet  with  con- 
centrated poles. 

Fig.  60  shows  the  field  about  a  concentrated  magnetic 
pole  in  an  infinite  medium  of  inductivity  ^  separated  by  a  plane 
interface  from  an  infinite  medium  of  inductivity  ^  for  the 
particular  case  in  which  A^//^  =  4,  the  field  within  the  mag- 
net being  neglected  and  its  opposite  pole  being  infinitely 
remote. 

Figs.  60,  61,  and  62  show  the  magnetic  field  in  and  around  a 
sphere  of  inductivity  ft2  placed  in  an  infinite  medium  of  inductivity 
ft1  supporting  an  originally  uniform  field  for  the  three  cases  in 
which  A^//4!  =  o,  3,  and  infinity. 


MAGNETS.     MAGNETOSTATIC    FIELDS.  281 

Fig.  63  shows  the  field  of  an  infinite  circular  cylindrical  shell 
of  inductivity  /JLZ  placed  in  an  infinite  medium  of  inductivity  /^ 
supporting  (originally)  a  uniform  magnetic  field  perpendicular  to 
the  axis  of  the  cylinders  for  the  particular  case  in  which  /^//^  = 
10.  §§14-15  were  in  fact  developed  largely  on  account  of  the 
magnetic  fields  exactly  analogous  to  the  electric  fields  there  in- 
vestigated. The  table  (Table  I.,  §§  14-15)  is  constructed  with 
values  of  c2/cl  or  /^//^  common  in  magnetism  but  never  occur- 
ring in  electrostatics.  The  use  of  spherical  and  cylindrical  iron 
shells  as  magnetic  screens  in  protecting  galvanometer  magnets 
from  external  fields  is  mentioned  in  §  30,  XII. 

Fig.  67  shows  the  field  of  a  uniformly  magnetised  isolated 
sphere,  Fig.  28,  with  the  additions  and  modifications  indicated  in 
§  6,  VI.,  that  of  an  infinite  circular  cylinder  uniformly  magnetised 
at  right  angles  to  its  length,  etc.,  etc.,  every  electric  field  having 
its  magnetic  analogue,  either  ideal  or  real. 

24.  Resultant  Magnetic  Poles,  Magnetic  Axis.  The  total  forcive 
upon  a  magnet  in  a  uniform  field,  as  the  earth's  magnetic  field, 
is  a  torque  tending  to  bring  its  axis  into  the  direction  of  the 
field.  It  is  evident  that  the  axis  of  the  magnet,  as  defined  in  §  4, 
is  the  direction  of  the  straight  line  drawn  from  the  center  of  the 
parallel  forces  due  to  the  uniform  field,  as  the  earth's  field,  on  all 
the  elements  of  the  distributed  negative  pole  to  the  center  of  the 
opposite  parallel  forces  upon  all  the  elements  of  the  positive 
pole. 

These  centers  may  be  regarded  as  the  positions  of  the  resultant 
poles  of  the  magnet ;  but  it  must  be  remembered  that  if  the 
forces  upon  all  the  elements  are  not  parallel,  as  they  are  when 
the  magnet  is  in  a  uniform  field,  the  positions  of  the  centers,  or 
points  of  application  of  the  resultant  forces,  are  different,  and 
different  for  every  different  field.  If  the  magnetism  were  uni- 
formly distributed  through  spheres  or  in  concentric  spherical 
shells,  or  over  spherical  surfaces,  at  the  two  ends  of  the  magnet, 
then  it  could  easily  be  shown,  as  in  the  corresponding  cases  in 


282          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

electrostatics  and  gravitation,  that  the  field  outside  the  spheres, 
and  the  resultant  forces,  would  in  all  cases  be  the  same  as  if  the 
poles  were  concentrated  at  the  centers  of  the  spheres.  The 
forces  between  the  poles  of  two  magnets  at  a  distance,  however, 
will  be  approximately  the  same  as  if  the  poles  were  concentrated 
in  the  positions  of  the  resultant  poles  for  a  uniform  field,  just  as 
the  gravitational  force  between  two  masses  of  irregular  shape, 
when  the  distance  between  them  is  considerable,  so  that  the  field 
due  to  each  is  nearly  uniform  at  the  other,  is  nearly  the  same  as 
if  the  masses  were  concentrated  at  their  centers  of  gravity  or 
centers  of  mass. 

For  convenience,  the  straight  line  drawn  from  the  negative  re- 
sultant pole  to  the  positive  resultant  pole,  as  well  as  its  direction, 
will  be  called  the  axis  of  the  magnet.  If  the  magnet  is  a  cylin- 
der of  symmetrical  cross-section  and  magnetised  symmetrically 
with  respect  to  its  geometrical  axis,  the  geometrical  and  magnetic 
axes  will  coincide. 

25.  The  Torque  Upon  a  Magnet  in  a  Uniform  Field,     Magnetic 
Moment.     Let  m  denote  the  magnitude  of  the  strength  of  each 
pole,  L  the  distance  between  the  resultant  poles,  H  the  intensity 
of  the  uniform  field,  and  6  the  angle  between  the  direction  of  the 
axis  and  the  direction  of  the  uniform  field.     The  torque  tending 
to  diminish  6  is  evidently  2.mH\L  sin  0,  and  the  torque  tending 
to  increase  0,  or  the  torque  measured  in  the   same  direction  as 
that  in  which  6  is  measured,  is 

T=  -  2mH\L  sin  d  =  -  mLH  sin  0  =  -  MH  sin  9    (33) 

if  we  put  mL  =  M. 

The  quantity  M=  mL  =  T/ff  sin  0  is  called  the  magnetic  mo- 
ment of  the  magnet  (analogous  to  the  electric  moment  of  an 
electret). 

26.  Gauss's  Method  of  Determining  Simultaneously  the  Moment 
M  of  a  Magnet  and  the  Horizontal  Component  H  of  the  Earth's 
Magnetic  Intensity.    I.   Determination  of  M  H .     The  magnet,  A, 
of  moment  Mis  first  suspended  with  its  axis  horizontal  by  a  ver- 


MAGNETS.     MAGNETOSTATIC    FIELDS.  283 

tical  fiber  as  free  from  torsion  as  possible,  and  set  to  vibrating 
through  a  very  small  horizontal  arc.  The  time  T  of  a  complete 
vibration  of  the  magnet,  and  its  moment  of  inertia  K  about  the 
axis  of  vibration,  are  then  determined.  Then  the  product  M  H 
is  given  by  the  equation 

Mtt  =  4T2K/r2  (34) 

to  a  high  degree  of  approximation. 

For,   since  T  =  Kd26  jdt2,  (33)  gives  for  the  equation  of  mo- 
tion of  the  magnet 

Kd26jdt2  +  M  H  sin  (9  =  o  (a) 

For   very    small    amplitudes  of  vibration   sin  6  =  6  very  ap- 
proximately and  (a)  becomes 

=  o  (b) 


which  shows  that  the  motion  is  simple  harmonic  in  the  period 
T=  27r(K/MH  )*,  a  relation  identical  with  (34). 

If  the  torsion  of  the  suspension  and  the  arc  of  vibration  are 
large  enough  to  have  appreciable  effects,  they  can  be  determined 
and  allowed  for  by  methods  given  in  Maxwell's  Treatise,  §§452 
and  738,  and  in  laboratory  manuals. 

II.  Determination  of  M/H.  The  magnet  A  is  removed  from 
the  suspension  and  mounted  with  its  axis  horizontal  and  per- 
pendicular to  the  magnetic  meridian  in  a  line  passing  through  the 
center  of  a  very  small  magnet  B  suspended  by  a  fiber  of  negli- 
gible torsion  (the  torsion,  if  not  negligible,  can  be  determined  and 
allowed  for),  the  whole  forming  a  magnetometer,  in  the  position 
occupied  by  the  center  of  A  in  I.,  and  the  distance  Rv  between 
the  centers  of  A  and  B  is  measured.  Rv  must  be  great  in  com- 
parison with  Z,  the  distance  between  the  resultant  poles  of  A, 
which  is  approximately  equal  to  the  length  of  the  magnet. 

The  magnetic  intensity,  Hlt  at  B  due  to  the  poles  of  A  is  per- 
pendicular to  H.  Hence  the  resultant  horizontal  intensity  at  B 
makes  with  the  magnetic  meridian  the  angle  6l  whose  tangent  is 

tantf.-^/H  (35) 


284         ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

The  axis  of  the  magnet  B,  in  the  meridian  before  the  introduc- 
tion of  A,  takes  up  a  position  parallel  to  the  resultant  intensity. 
That  is,  on  the  introduction  of  A,  B  is  deflected  through  the 
angle  Ov  which  must  be  measured. 

Denoting  the  numerical  strength  of  each  pole  of  A  by  m  and 
assuming  the  magnetisation  of  A  to  be  symmetrical,  so  that  the 
distance  of  each  resultant  pole  is  LJ2  from  the  center  of  A,  we 
have  for  H 


•    i  4- 
=  M/2irpR*  •  (i  +  L2/2R2)  =  H  tan  0l  (c) 

if  the  fourth  and  higher  powers  of  LjRl  are  neglected. 

Since  (c)  contains  the  unknown  quantity  L  in  addition  to  M 
and  H,  the  experiment  is  repeated  with  a  different  distance,  R2, 
between  the  centers  of  A  and  B,  and  the  corresponding  angle  of 
deflection  #2  is  determined.  Then 


*  •  (i  +  £2/2R2)  =  H  tan  02  (d) 

Eliminating  L  from  (c)  and  (//),  we  obtain 

M/H  =  nrn(R*  tan  0l  -  X'  tan  ^2)/(^2  -  Xfl          (36) 


From  (34)  and  (36)  both  J/and  H  can  be  calculated,  since  //. 
for  air  is  known  and  sensibly  equal  to  ytt0  =  i  . 

To  eliminate  the  error  arising  from  the  (possible)  lack  of  sym- 
metry in  the  magnetisation  of  A,  and  errors  in  determining  6V 
02,  Rv  and  R2,  the  angles  are  read  for  each  value  of  R,  with  A 
either  east  or  west  of  B,  first  with  the  one  pole  of  A  toward  B, 
then  with  the  other  pole  toward  B.  Then  A  is  placed  on  the 
opposite  side  of  B  and  the  angles  read,  for  both  directions  of  the 
axis  of  A,  with  the  same  values  of  ^  and  R2.  The  mean 
values  of  6l  and  02  derived  from  all  the  readings  are  used  in  the 
calculation  of  M\\\  by  (36). 


MAGNETS.     MAGNETOSTATIC   FIELDS.  285 

27.  The  Comparison  of  Two  Intensities.  By  vibrating  the  same 
magnet  successively  in  two  fields  of  intensities  Hl  and  Hv  we 
obtain  from  (34), 

(37) 


By  using  the  same  magnet  A  to  deflect  a  small  magnet  B  in 
two  different  fields  of  intensities  H^  and  Hv  A  being  placed  with 
reference  to  the  center  of  B  and  the  direction  of  the  field  in  each 
case  as  described  in  II.,  and  the  distance  between  the  centers  of 
the  magnets  being  the  same  in  both  cases,  we  obtain  from  (35) 

tan  ejtomO^HJH^  (38) 

The  Comparison  of  Two  Magnetic  Moments.  By  vibrating  in 
the  same  field  two  magnets  of  moments  Ml  and  Mv  and  moments 
of  inertia  about  the  axis  of  suspension  K^  and  Kv  we  obtain 
from  (34) 

?  (39) 


The  comparison  may  also  be  made  by  means  of  equation  (36). 


CHAPTER    XII. 
THE   MAGNETIC    FIELD    OF   THE    CONDUCTION    CURRENT. 

1.  Relation  Between  the  Direction  of  a  Current  and  the  Direc- 
tion of  its  Lines  of  Intensity,  An  electric  conduction  current  is 
invariably  accompanied  by  a  magnetic  field  surrounding  and 
penetrating  into  the  conductor.  In  the  case  of  a  long  straight 
circular  cylindrical  wire  carrying  a  current  and  immersed  in  a 
medium  of  uniform  inductivity  the  lines  of  magnetic  intensity,  as 
will  be  shown  in  §  15,  are  circles  centered  on  the  axis  of  the 
wire  in  planes  perpendicular  to  the  axis,  and  the  direction  of  each 
line  is  related  to  the  direction  of  the  current  as  the  rotation  to 
the  translation  of  a  right-handed  screw  or  as  the  direction  of 


Line  of  Magnetic 
Intensity 


Current 


Fig.  84. 

motion  of  the  hands  of  a  clock  to  the  direction  of  a  line  drawn 
from  the  face  to  the  back.  If  for  circle  the  words  closed  curve 
are  substituted,  these  statements  hold  good  in  all  cases.  The 
relation  between  the  direction  of  a  current  and  that  of  its  lines 
of  intensity  is  shown  in  Fig.  84. 

2.  Positive    Directions  Around    and  Through  a  Circuit.     The 

positive  direction  around  a  circuit  bears  to  the  positive  direction 
through  the  circuit,  by  definition,  the  same  relation  as  the  direc- 
tion of  rotation  of  a  right-handed  screw  bears  to  its  direction  of 

286 


CURRENTS  AND  MAGNETIC    FIELDS. 


287 


translation.  Thus  if  AB,  Fig.  85,  is  chosen  as  the  positive  di- 
rection through  the  circuit  CDEC,  the  positive  direction  around 
the  circuit  is  CDEC  as  indicated  by  the  arrows.  If  the  direction 

r\ 


Fig.  85. 

of  the  current  in  a  circuit  is  chosen  as  the  positive  direction 
around  the  circuit,  as  will  be  done  in  this  chapter,  the  magnetic 
flux  connected  with  the  current  will  always  thread  the  circuit  in 
the  positive  direction. 

3.  Vector  Product  of  Two  Vectors.  Let  A  and  B  denote  two 
vectors  intersecting  at  an  angle  6  less  than  TT.  If  Ct  a  third 
vector,  is  equal  in  magnitude  to  the  product  AB  sin  0,  and  is 
perpendicular  to  their  plane  and  so  directed  that  a  right-handed 


-VAB  sin 

,=VBA  sin  0 


Fig,  86. 


screw  progressing  along  C  in  the  positive  direction  would  rotate 
from  A  to  B,  Fig.  86,  then  C  is  called  the  vector  product  of  A 
and  B,  and  is  written 

C  =  VAB  sine  (i) 


288          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

If  C  has  the  opposite  direction,  as  indicated  by  the  dotted  line 
in  the  figure,  it  may,  consistently  with  (i),  be  written 


C=  -  NAB  sin  0  =  MBA 


sn 


4.  The  Law  of  Ampere,  If  a  very  long  and  thin  cylindrical 
conductor  carrying  a  current  is  immersed  in  a  uniform  magnetic 
field  perpendicular  to  its  direction,  the  two  fields  will  combine  to 
form  the  resultant  field  shown  in  cross-section  in  Fig.  87,  the 
resultant  field  being  stronger  where  the  component  intensities 
have  the  same  general  direction,  and  weaker  where  their  gen- 
eral directions  are  opposite.  The  pressures  and  tensions  in  the 
field  will  then  give  rise  to  a  force  upon  the  conductor  directed 
from  the  stronger  to  the  weaker  part  of  the  resultant  field,  and 
thus,  as  the  figure  shows,  perpendicular  to  the  wire  and  to  the 
originally  uniform  field.  In  the  figure  the  conductor,  perpen- 
dicular to  the  paper,  is  shown  as  a  small  circle  ;  the  current 
flows  downward  and  the  original  field  is  directed  to  the  left,  or 
the  current  flows  upward  and  the  original  field  is  directed  to  the 
right.  In  either  case  the  force  upon  the  conductor  is  directed 
toward  the  top  of  the  page.  This  is  a  simple  case  coming 
under  the  law  of  Ampere,  to  which  we  proceed. 

Consider  a  wire  of  negligible  cross  -section  carrying  a  current  / 
in  a  magnetic  field  of  any  kind.  Let  dL  denote  the  element  of 
length  of  the  wire  at  any  point  P,  and  let  B  denote  the  magnetic 
induction  at  the  point*  Let  6  denote  the  angle  between  /  and 
B.  (For  convenience  we  shall  here  treat  /  as  if  it,  as  well  as  the 
current  density  i,  of  which  it  is  the  surface  integral,  were  a  vec- 
tor.) Then  the  results  of  experiment  may  be  summed  up  in  the 
following  statement,  which  constitutes  Ampere's  law  in  the  first 
form  : 

The  wire  is  acted  upon  by  a  force  F  or  a  torque  T  which  may 
be  found  by  assuming  each  element  of  the  wire  of  length  dL  to 
be  acted  upon  by  a  force 

dF=  i  {a-  V  IB  sin  0    dL  (2) 

*  See  Lord  Rayleigh,  Philosophical  Magazine,  June,  1898. 


CURRENTS  AND  MAGNETIC  FIELDS.  289 

where  a  is  a  constant  depending  on  the  units  in  which  the  other 
quantities  in  (2)  are  expressed,  and  taking  the  vector  summation 
or  integration  of  all  the  elementary  forces  dF  along  the  part  of 


Fig.  87. 


the  wire  considered,  or  of  all  the  elementary  torques  dT  arising 
from  the  forces  dF.  The  force  upon  dL  is  always  from  the 
stronger  to  the  weaker  part  of  the  resultant  field. 


290          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

If  the  wire  is  not  so  thin  that  its  cross-section  can  be  neg- 
lected, it  may  be  considered  split  up  into  elementary  current 
tubes  and  the  summation  or  integration  applied  to  all  the  ele- 
ments of  each  tube. 

5,  The  Rational  Electromagnetic  Unit  Current.  If  in  the  above 
equation  dF  and  dL  are  expressed  in  c.g.s.  units,  /  in  the  RES 
unit,  and  B  in  the  REM  unit,  a  is  found  to  have  the  magnitude 
3  x  io10  (see  §4,  XIV.).  The  REM  unit  current  is  defined  as 
a  unit  current  a  times  as  great  as  the  RES  unit  current.  Hence, 
if  we  express  /  in  terms  of  this  unit,  I  la  [if  a  is  assumed  to  have 
zero  dimensions  (XIV.)]  will  disappear  from  the  above  equation, 

which  thus  becomes 

dF=VIB  sin  6  •  dL  (3) 

(?  "fc^  Electric  Units  in  the  Rational  Electromagnetic  Unit  System. 
If  the  electric  current  is  expressed  in  terms  of  the  electromag- 
netic unit,  however,  the  relations  q  =  //,  dHjdt=RI2,  ¥  =  RIt 
etc.,  will  not  remain  true  unless  the  units  of  charge,  resistance, 
e.m.f.,  etc.,  are  redefined.  Units  so  chosen  as  to  make  these  re- 
lations, or  any  of  the  relations  which  precede  or  follow,  correct 
when  /,  or  any  other  quantity  occurring  in  the  relations,  is  ex- 
pressed in  the  REM  unit  current  are  defined  as  the  REM  unit 
charge,  resistance,  etc. 

Magnetic  and  Electromagnetic  Units  in  the  Rational  Electro- 
static System.  We  have  not  hitherto  defined  any  pure  magnetic 
unit,  such  as  magnetic  pole  strength  or  magnetic  intensity,  in  the 
electrostatic  system  ;  but  from  the  above  definitions  of  the  REM 
unit  current,  charge,  etc.,  in  terms  of  the  RES  units  we  can  pro- 
ceed immediately  to  such  definition  :  Units  so  chosen  as  to  make 
(3)  and  all  the  equations  which  follow  in  this  work,  as  well  as  all 
those  of  Chapter  XL,  correct  when  7,  or  any  other  quantity 
occurring  in  the  equations,  is  expressed  in  the  RES  unit  are 
defined  to  be  the  rational  electrostatic  units  of  the  quantities 
concerned. 


CURRENTS  AND  MAGNETIC    FIELDS. 


29I 


Henceforth  every  equation  will  be  expressed  in  one  system  of 
units  throughout,  like  all  preceding  equations  except  (2),  all  the 
quantities  being  expressed  in  RES  units,  or  else  all  in  REM units. 

The  general  subject  of  electric  units  is  discussed  in  Chap.  XIV. 

7.  The  Force  Upon  a  Straight  Wire  in  a  Uniform  Magnetic  Field. 

As  a  particular  case  falling  under  Ampere's  law,  we  shall  con- 
sider first  a  straight  wire  in  a  uniform  magnetic  field,  and  shall 
find  the  force  upon  a  length  L  of  the  wire. 

(1)  If  the  wire  is  parallel  to  B,  sin  6  —  o  everywhere,  hence 
F=o. 

(2)  If  the  wire  is  perpendicular  to  the  field,  sin  0  =  i  every- 
where, and  F=  IBL  ^ 

The  resultant  field  (a  uniform  field  and  the  field  of  §  14  super- 
posed) and  the  direction  of  F  are  shown  in  Fig.  87  (from  Max- 
well's Treatise,  §  496). 

(3)  If  the  wire  makes  an  angle  6  with  B 

F=  IB  sin  6L  =IBL  sin  0  (5) 

The  force  is  the  same,  both  in  magnitude  and  direction,  as 
would  be  exerted  on  the  wire  in  a  field  B  sin  6  perpendicular  to 
L,  or  upon  a  wire  of  length  L  sin  6  in  a  field  perpendicular  thereto. 

8.  A  Linear  Circular  Circuit  in  the  Radial  Field  from  a  Concen- 
trated Magnetic  Pole  of  Strength  m  Placed  (1)  at  its  Center.     (Fig. 
88.)     The  magnetic  induction  and  the  electric  current  in  the  lin- 


Currertt  down 


Force  on  Olrcutl 


Fig.  88. 


292 


ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 


ear  circuit  are  obviously  everywhere  perpendicular.     The  force 
upon  the  circuit  is 

F  =  if  B  sin  0  dL  =  I(m  /  47rR2)27rR  =  mI/2R  (6) 

The  direction  of  the  force  is  downward  into  the  paper  in  Fig. 
88«  and  to  the  right  in  Fig.  88$,  when  the  current  has  the  direc- 
tion indicated  and  m  is  positive.  If  the  sign  of  either  is  changed, 
the  direction  of  /MS  reversed. 

(2)  On  the  Axis  of  the  Circle  at  the  Distance  d  from  its  Center. 
If  the  pole  is  placed  on  the  axis  of  the  circle  at  the  distance  d 


Fig.  89. 

from  its  center,  Fig.  89,  B  sin  0  =  B  =  mj^d2  +  R2),  and  the 
force  upon  each  element  of  length  dL  of  the  wire  is 

dF=  IB  sin  OdL  =  Im/47r(d2  +  R2)  •  dL 
This  force  may  be  resolved  into  two  components,  one 

dF'  =  dF  sin  a  =  mlRj^d2  +  R2)*  •  dL 
in  the  direction  of  the  axis  ;  and  the  other 


in  the  direction  of  the  radius.  The  second  component  tends  to 
increase  or  diminish  the  radius  of  the  wire  according  as  the  flux 
from  the  pole  threads  the  circuit  in  the  positive  or  negative  direc- 
tion through  the  circuit,  but  gives  rise  to  no  resultant  force  upon 
the  circuit  as  a  whole  in  any  direction.  The  first  component, 


CURRENTS  AND    MAGNETIC    FIELDS. 


293 


summed  up  for  the  whole  length  of  the  wire,  gives,  for  the  total 
force  upon  the  wire  in  the  direction  of  the  axis, 

F=  f  dFf  =  mIR2/2(d2  +  R2)*.  (7) 

When  d  =  o,  (7)  reduces  to  (6). 

9,  An  Infinite  Straight  Linear  Wire  in  the  Radial  Field  from  a 
Concentrated  Magnetic  Pole  of  Strength  m  Distant  d  from  the  Wire. 
(Fig.  90.)  In  this  case  F  has  the  magnitude 


Fig.  90. 

F=  I$B sin 0 dL=  Imj^ -fi/(L2  +  d2)  •  dj(L2  +  d2)* •  dL 
=  Imdl2ir  •  fi  /(L2  +  d2)*  -  dL  =  Im^ird  -  f^  sin  0  dQ    (8) 


since  L  =  d  cotan  0. 

The  relative  directions  of  Ft  B,  and  the  current  are  obvious. 

10.  A  Closed  Plane  Circuit  of  Any  Form  in  a  Uniform  Magnetic 
Field.  (Fig.  91.)  Let  the  plane  of  the  circuit  be  vertical  and 
the  field  horizontal.  Let  zt  x,  be  the  coordinates  of  any  point 
P  of  the  circuit,  referred  to  its  highest  point  0  as  origin,  the 
positive  direction  of  Z  being  taken  as  the  direction  of  the  current 
at  O,  and  the  positive  direction  of  X  vertically  downward.  Let 
a  denote  the  angle  between  B  and  the  positive  direction  of  the 
Z  axis. 


294 


ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 


Let  B  be  resolved  into  two  components  B  sin  a  and  B  cos  a 
perpendicular  and  parallel,  respectively,  to  the  plane  of  the  circuit. 

Owing  to  the  component  B  sin  a,  there  will  be  an  outward 
force  dFxz  —  IB  sin  a  dL  upon  every  element  of  the  circuit  per- 


.  91. 


pendicular  to  its  length  dL  and  to  B  sin  a  in  the  plane  of  the 
circuit.  Resolving  this  force  along  OX  and  OZ>  we  have,  if  dL 
(taken  in  the  direction  of  the  current)  makes  an  angle  /3  with  OX, 

dFe  —  dFxtt  cos  /3  =  IB  sin  a  cos  /3  dL  =  IB  sin  a  dx 
and 

dFx  =  dFxn  cos  (90°  +  /3)  =  -  IB  sin  asm  &dL=  —  IB  smadz 
By  integration  along  the  whole  circuit 

Fn  =  IB  sin  a.$dx  =  o 

and  Fx  =  —  IB  sin  a  fdz  =  o 

Hence  the  forcivc  due  to  B  sin  a,  if  not  zero,  is  a  torque.     Since 


CURRENTS  AND    MAGNETIC    FIELDS.  295 

all  the  forces  dFxs  are  in  the  plane  of  the  circuit,  there  can  be  no 
torque  about  an  axis  in  this  plane.  Let  therefore  O  Y  (the  posi- 
tive direction  through  the  circuit),  perpendicular  to  6Mfand  OZ 
at  0,  be  taken  as  the  axis  of  moments.  The  force  dFxz  on  the 
element  dL  gives  rise  to  a  torque 

dT  =  dFxz  —  dFzx  =  —  IB  sin  a  zdz  —  IB  sin  a.  xdx 
=  —  IB  sin  a  (zdz  +  xdx] 

By  integration  around  the  circuit  we  obtain 

T=  §dT  =  —  IB  sin  a.  (§zdz  +fxdx)  =  o 

Hence,  so  far  as  the  component  B  sin  a.  is  concerned,  there  is 
no  resultant  torque  or  force  upon  the  whole  circuit.  Under  the 
action  of  the  force  dFxx  —  IB  sin  a  dL,  however,  each  element  of 
the  circuit  tends  to  move  outward  or  inward  according  as  sin  a 
is  positive  or  negative,  i.  e.,  according  as  the  flux  of  the  uniform 
field  threads  the  circuit  in  the  positive  of  negative  direction 
through  the  circuit ;  thus  the  circuit,  if  made  of  elastic  material, 
would  expand  or  contract. 

The  component  B  cos  a  gives  rise  to  a  force  dFy  on  the  ele- 
ment dL  perpendicular  to  the  plane  XZ  of  the  circuit.  If  O  Y  is 
taken  as  the  direction  of  a  positive  force, 

dF  =  —  IB  cos  a  cos  /3  dL  =  —  IB  cos  a  dx 
and  the  total  force  upon  the  circuit  in  the  direction  OYis 

F  =  CdF  =  -  IB  cos  a  fdx  =  o 

y       \)        y  u 

Hence  the  forcive,  if  anything,  is  a  torque.  Since  there  is  no 
force  on  any  element  in  the  direction  OZ  or  Z0t  there  can  be  no 
torque  about  OY.  The  torque  about  OZ  is 

Tg  =fdT,  =$xdFy  =  —$xIB  cos  a  dx  =  —  IB  cos  afxdx  =  O 
The  torque  about  the  axis  OX  is 

Tx  =fdTx  =  -fsdFy  =  IB  cos  afzdx 

=  IB  cos  a  fdS  =  IB  cos  a.  S 
if  vS  denotes  the  area  of  the  circuit. 


296          ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

Thus  the  total  forcive  upon  the  circuit  is  a  pure  torque 

T=f£cosaS  (9) 

around  a  line  in  the  plane  of  the  circuit  and  perpendicular  to  the 
original  magnetic  field,  the  positive  direction  of  the  torque  being 
always  related  to  the  positive  direction  of  OX  as  the  rotation  to 


Current  down 


\C* *^v   Torque 


Current  up 


Fig.  92. 

the  translation  of  a  right-handed  screw  with  OX  as  axis.     When 
the  torque  is  positive  it  tends  to  increase  the  angle  a  (see  Fig. 

93). 

If  there  are  n  turns  in  the  circuit,  the  torque  is  ;/  times  as 
great. 

If  0  denotes  the  angle  between  the  normal  to  the  circuit  (in  the 
positive  direction,  §  2)  and  B,  the  torque  tending  to  increase  the 
angle  6  is  T==  _  JBS  ^Q=_  IflSH  sin  0  (lo) 

so  that  the  circuit  behaves  like  a  magnet  of  moment  7/^5. 

When  0  =  o°,  T—  o,  and  the  circuit  is  in  stable  equilibrium  ; 
and  when  6=  180°,  T=o,  and  the  circuit  is  in  unstable  equil- 
ibrium. For  in  the  former  position  the  torque  brought  into 
existence  by  a  slight  change  of  6  will  tend  to  restore  equilibrium  ; 
while  in  the  latter  the  torque  developed  by  a  similar  change  in 
0  will  tend  to  increase  the  displacement. 

When  0  =  o  and  T=o,  the  circuit  encloses  the  maximum 
magnetic  flux  possible,  a  quantity  US  due  to  the  uniform  field, 


CURRENTS  AND  MAGNETIC    FIELDS.  297 

and  that  connected  with  its  own  current ;  the  uniform  flux,  in 
this  position  of  the  circuit,  being  directed  through  the  circuit  in 
the  positive  direction. 

When  6  =  90°  or  270°,  the  circuit  encloses  no  flux  of  the 
uniform  field,  and  the  torque  is  a  maximum. 

The  circuit  thus  tends  to  move  in  such  a  way  as  to  enclose 
the  maximum  flux  possible. 

We  have  already  noted  the  tendency  of  a  closed  circuit  carry- 
ing a  current  to  expand  or  contract  when  placed  in  a  radial  field 
or  a  uniform  field,  according  as  the  magnetic  flux  of  this  field 
threads  the  circuit  in  the  positive  or  negative  direction  through 
the  circuit.  That  is,  the  circuit  expands  or  contracts  according 
as  the  expansion  or  contraction  will  increase  the  flux  through 
the  circuit  in  the  positive  direction,  or  diminish  the  flux  through 
the  circuit  in  the  negative  direction.  For  the  same  reason 
(Ampere's  law)  the  circuit  would  tend  to  expand  if  only  in  its 
own  field,  and  thus  to  enclose  as  great  a  quantity  of  magnetic 
flux  as  possible. 

Similar  considerations  would  show  that  in  every  case  a  circuit 
carrying  a  current  moves  or  tends  to  move  in  such  a  manner  as 
to  make  the  magnetic  flux  threading  it  as  great  as  possible. 

11,  Magnetic  Intensity  Due  to  any  Current  Distribution. — From 

Ampere's  law  (first  form)  and  the  third  law  of  motion  we  can 
find,  for  any  point  P,  an  expression  for  the  magnetic  intensity 
connected  with  any  current  distribution  (Ampere's  law,  second 
form). 

To  do  this  imagine  the  conductor  immersed  in  a  radial  field 
from  a  concentrated  magnetic  pole  of  strength  m  at  the  given 
point  P  distant  r  from  dL,  an  element  of  length  of  the  wire  or 
current  tube  under  consideration.  Then  the  induction  at  dL  due 
to  the  pole  is  B  =  m/^r2  directed  radially  from  P.  If  we  write  r 
in  the  numerator  to  indicate  the  direction  of  Bt  r  (=  r  numerically) 
being  measured  from  the  pole  to  dL,  this  expression  becomes 

B  = 


298          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

The  force  upon  dL  is  therefore 

dF  =  mf/47rr3  -  V  dLr  sin  6 

for  a  linear  current,  dL  being   measured  in  the  direction  of  the 
current.      Hence 

F  =  ml/4.7r  X  vector  integral  of  I  /r3-  VdLr  sin  0  jr* 

Since  F  is  the  total  force  upon  the  circuit,  there  must  be  an 
equal  and  opposite  force,  —  Fy  upon  the  pole  at  P.  Hence,  if  H 
denotes  the  magnetic  intensity  at  P  due  to  the  current, 


Current  Tube 


Fig.  93. 


—  F=Hm  =ml/4.7r  x  vector  integral  of  (VrdL  sin  O) 
or,  if  r  is  measured  from  dL  to  the  point  Py  instead  of  from  P  to  dL, 
Hm  =  ml/47r  x  vector  integral  of  (V  'dLr  sin 


whence 

H—II^TT  x  vector  integral  of  (VdLr  sin  6)/r3  (i  i) 

The  magnetic  intensity  at  P  is  thus  the  same  as  if  each  ele- 
ment of  the  current  (IdL)  produced  at  P  an  intensity 

dH  =  //47rr  3  •  MdL  r  sin  6  (i  2} 


CURRENTS    AND    MAGNETIC    FIELDS.  299 

magnitude  and  direction  (r  being  measured  from  dL  to  P)  ;  or, 
numerically,  7/4?rr2  •  sin  6  dL. 

If  the  conductor  has   not  a  negligible  cross-section,  we  have, 

clearly, 

(13) 


and  H—Ij^irr^-  vector  integral  (VzV  sin  6}dr  (14) 

Equations  (i  i)-(i4)  express  Ampere's  law  in  its  second  form. 

It  must  be  remembered  that  the  integral  force  F  and  the  inte- 
gral intensity  H,  not  the  elements  dF  and  dH,  are  all  that  experi- 
ment furnishes. 

The  relations  between  the  quantities  of  equation  (13)  are 
shown  in  Fig.  93.  dL,  2,  and  r,  are  taken  in  the  XY  plane,  and 
the  direction  of  PXt  perpendicular  to  the  XY  plane,  is  chosen  to 
coincide  with  that  of  i  and  dL. 

If  i  (or  7)  has  everywhere  one  direction,  then  it  is  obvious 
from  (12)  or  (13),  that  there  is  no  component  of  magnetic  in- 
tensity in  this  direction  anywhere. 

From  all  the  above  equations  it  is  manifest  that  H  does  not 
depend  in  any  way  upon  the  inductivity  of  the  medium  in  which 
the  conductors  are  placed,  provided  that  the  medium  is  homo- 
geneous and  isotropic  (so  that  the  field  from  a  concentrated  pole 
anywhere  is  radial). 

In  what  follows  all  media  will  be  supposed  homogeneous  and 
isotropic  unless  the  contrary  is  stated. 

12.  The  Magnetic  Field  Around  an  Infinitely  Long  Straight 
Linear  Conductor  Carrying  a  Current  I.  From  equation  (12),  or 
from  the  direct  application  of  the  third  law  of  motion  to  (8),  §  9, 
it  is  evident  that  the  lines  of  intensity  are  circles  centered  on  the 
wire  and  perpendicular  thereto,  the  direction  of  the  lines  around 
the  wire  being  related  to  the.  direction  of  the  current  as  the  rota- 
tion to  the  translation  of  a  right-handed  screw.  The  magnitude 
of  the  intensity  is,  by  (i  i)  or  (8), 

H=F/m  =  I/2ird  (15) 

at  a  distance  d  from  the  wire. 


300          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

Maxwell's  plane  diagram  of  a  part  of  the  field  is  shown  in  Fig, 
1 6,  Chapter  II. 


Fig.  94. 


13.  The  Magnetic  Intensity  at  a  Point  on  the  Axis  of  a  Circular 
Linear  Conductor  Carrying  a  Current  I.     From  (12),  or,  more 


CURRENTS   AND    MAGNETIC    FIELDS. 


3O1 


readily,  from  the  direct  application  of  the  third  law  of  motion  to 
(7),  §  8,  the  intensity  at  a  point  on  the  axis  distant  d  from  the 
center  of  the  circle  is,  numerically, 

H=  F/m  =  IR2/2(R2  +  d^  (16) 

and  is  directed  along  the  axis  in  the  positive  direction  through 
the  circuit. 

If  the  circular  circuit  contains  n  turns,  closely  wound,  instead 

of  one>  H=  nIR2l2(R2  +  d^  (17) 

and  if  there  are  two  similar  coils  at  the  same  distance  d  on  oppo- 
site sides  of  P,  with  their  currents  of  the  same  magnitude  and  in 
the  same  direction,  H=  n/j^^  +  ^  (18) 


Fig.  95. 

Putting  d  =  o,  we  obtain  for  the  intensity  at  the  center  of  a 
circular  coil  of  n  turns,       //= 


Maxwell's  diagram  of  the  field  connected  with  a  single  coil  is 
given  in  Fig.  94,  and  that  of  the  field  connected  with  two  similar 


302          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

parallel  coils  carrying  the  same  current  in  the  same  direction  in 
Fig.  95  (see  Maxwell's  Treatise,  §§487,  702,  and  713,  from 
which  the  figures  are  taken). 

14.  The  M.M.F.  Around  an  Infinite  Linear  Straight  Wire  Carrying 
a  Current  I.  The  magnetomotive  force  along  the  arc  P^P2  from 
Pl  to  Pv  two  points  on  the  same  line  of  intensity  of  radius  d,  Fig. 

96'  1S  NX  arc  />/>  =  /  arc  PlP2J2-jrd  =  I  0  j  2-rr  (20) 

where  6  denotes  the  angle  subtended  at  the  wire  by  the  arc  P^P2. 

The  equation  shows  that  the  m.m.f.  along  the  arc  of  any  line 

of  intensity  from  a  plane  ABE  to  a  plane  CBE,  intersecting  in 


Fig.  96. 

the  wire  at  the  angle  0,  is  equal  to  I  0/2ir,  independently  of  the 
value  of  d. 

The  m.m.f,  moreover,  along  any  path  not  enclosing  the  cir- 
cuit, as  P^KP^  from  Pv  any  point  in  the  plane  ABE,  to  Py  any 
point  in  the  plane  CBE,  is  the  same,  and  equal  to  the  value  given 
in  (20).  For  the  line  integration  of  the  intensity  along  any  path 
is  compounded  of  integrations  along  lines  of  intensity,  integrations 
along  radii,  and  integrations  parallel  to  the  wire.  The  integrals 
in  the  last  two  directions  are  zero,  since  they  are  perpendicular 
to  the  intensity,  and  the  total  integral  along  the  lines  of  intensity 
is  as  before,  I  dJ2ir. 

The  m.m.f.  between  two  points  Pl  and  P2  therefore  depends 
only  on  the  strength  of  the  current  /  and  the  fraction  of  a 
circumference  6 /2?r  traversed  in  passing  from  Pl  to  P2  or  from 
PI  to  Pv 

Thus  the  m.m.f.  along  a  closed  circuit  not  enclosing  the  cur- 
rent, as  the  line  P^KPff^  is  zero,  no  fraction  of  a  circumference 


CURRENTS   AND    MAGNETIC    FIELDS.  303 

being  traversed,  on  the  whole,  in  passing  from  Pl  to  Pl  again. 
This  may  be  shown  also  as  follows  :  The  m.m.f.  along  the  path 
PfiPfi  is  /  0/27T,  and  the  m.m.f.  along  Pf^  is  -  f0/2ir.  Hence 
the  total  m.m.f.  is  zero. 

The  m.m.f.   around  a  closed  path   linking  with  the  current 

a=/27T/27T=/  (21) 

the  directions  of  the  m.m.f.  and  current  being  related  like  the 
rotation  and  translation  of  a  right-handed  screw.  If  the  closed 
path  links  n  times  with  the  current,  the  m.m.f.  is 

O  =  nl  (22) 

This  proposition  will  be  generalised  in  Chapters  XIII.  and  XV. 
It  will  there  be  shown  that  the  m.m.f.  along  any  closed  path 
linking  n  times  with  any  current  is  equal  to  n  times  the  current. 
The  relation  is  known  as  the  first  law  of  circulation,  and  will  be 
assumed  in  what  follows. 

15.  The  Magnetic  Field  of  an  Infinitely  Long  Cylindrical 
Homogeneous  Conductor  of  Circular  Cross-section.  According  to 
§11,  Chapter  VIII.,  the  current  density,  z,  is  uniform  throughout 
the  conductor  and  has  the  direction  of  its  axis.  Hence  it  follows 
from  the  symmetry  of  the  conductor  and  equation  (13)  that  the 
intensity  at  any  point  has  no  component  in  the  direction  of  the 
radius  or  axis  of  the  conductor.  The  lines  of  intensity  are  there- 
fore circles  centered  on  the  axis  in  planes  perpendicular  thereto. 
The  intensity  at  a  point  distant  d  from  the  axis  of  the  conductor 
can  now  be  determined  by  applying  (21). 

Outside  the  conductor,  that  is,  for  a  circle  of  radius  d  greater 
than  R,  the  radius  of  the  conductor,  we  have,  for  the  m.m.f. 
around  the  circle  in  the  direction  of  the  intensity, 


from  which 

(23) 


just  as  for  a  linear  wire  carrying  the  same  current. 


304          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

Inside  the  conductor,  that  is,  for  a  circle  of  radius  d  less  than  Rt 

H  =  H2ird  =  iird2  =  Id2  / '  R2 
and  H=  idJ2  =  lrd/27rX2  (24) 

At  the  surface  of  the  conductor,  where  d—  R,  (23)  and  (24) 
become  identical  as 

H=iRJ2  (25) 

At  the  axis  of  the  conductor,  where  d=  o,  (24)  gives  H  =  o. 

Maxwell's  plane  diagram  of  the  magnetic  field  within  and  with- 
out the  conductor  is  easily  drawn  by  the  method  developed  in 
Chapter  II.  The  lines  of  intensity  and  equipotential  lines  outside 
the  wire  are  exactly  similar  to  the  equipotential  lines  and  lines 
of  intensity,  respectively,  of  the  diagram  of  the  electrostatic  field 
of  §  II,  Chapter  II.  The  development  of  the  formulae  for  draw- 
ing the  lines  of  intensity  so  that  the  spaces  between  successive 
lines  correspond  to  tubes  of  equal  strength  is  left  to  the  reader. 

If  the  same  current  flows  along  a  homogeneous  circular  cylin- 
drical shell  of  infinite  length,  the  field  outside  the  conductor  is 
the  same  as  that  given  by  (23),  but  inside  the  intensity  gradually 
diminishes  until  it  vanishes  at  the  inner  surface  of  the  shell. 
Within  the  space  inclosed  by  the  inner  surface  of  the  shell  there 
is  no  field.  For  if  there  were  a  field,  it  would  be  circular  and 
perpendicular  to  the  axis,  and  the  intensity  at  the  distance  d  from 
the  axis  would  be  given  by  (24).  Since  in  this  region  i  =  o,  H 
is  also  zero. 

The  same  method  may  be  applied  to  the  case  of  a  conductor 
consisting  of  any  number  of  coaxial  circular  shells,  each  infinite  in 
length  and  homogeneous.  The  external  field  is  the  same  as  that 
which  would  surround  a  linear  wire  at  the  axis  carrying  the 
same  current. 

16.  A  Tore  of  Inductivity  /i2  Symmetrically  Placed  in  a  Circu- 
lar Field  in  a  Medium  of  Inductivity  /*lf  etc.  If  a  tore,  or  circu- 
lar ring  of  constant  cross-section,  of  inductivity  /x2is  placed  around 
the  cylindrical  conductor  or  any  of  the  conductors  of  §§  14-1 5  im- 


CURRENTS   AND    MAGNETIC    FIELDS.  305 

mersed  in  a  medium  of  inductivity  ^  with  its  axis  coincident  with 
that  of  the  cylinder  (or  other  conductor),  the  field  external  to  the 
tore  will  remain  unaffected,  except  during  the  introduction  of  the 
tore,  and  the  intensity  within  the  tore  will  also  remain  unaffected, 
by  the  principle  of  symmetry  and  equations  (17)  XL,  and  (21). 
but  the  induction  at  any  point  within  the  tore  will  be  increased 
by  its  introduction  in  the  ratio  /*2/Air 

Since  the  magnetic  pressure  perpendicular  to  the  intensity  and 
to  the  interface  at  a  point  P  just  outside  the  tore  in  medium  I  is 
J/-^//2  and  the  pressure  just  within  the  tore  an  infinitesimal  dis- 
tance from  P,  J/-t2//2,  there  will  be  a  normal  mechanical  force 
upon  the  interface  equal,  per  unit  area,  to 


T=  K//2  -  ^ff2  =  \H\^  -  /»,)  (26) 

if  considered  positive  when  directed  from  medium  2  to  medium 
I.  Thus  if  fa  is  greater  than  pv  the  volume  of  the  tore  will  in- 
crease slightly  on  its  introduction  into  the  field. 

This  case  and  that  of  §  7,  IV.,  are  the  limiting  cases  of 
§  9,  IV. 

In  exactly  the  same  way  the  matter  within  any  (closed)  tube 
of  induction  in  a  homogeneous  medium  of  inductivity  ^  may  be 
replaced  by  a  substance  of  inductivity  /n2.  The  external  field,  as 
well  as  the  intensity  within  the  tube,  will  not  be  disturbed  ;  but 
the  induction  within  the  tube  will  be  increased  in  the  ratio  p2/f*lt 
and  a  normal  traction  will  be  developed  at  each  point  of  the  in- 
terface. If  H  denotes  the  intensity  at  any  point  of  the  interface, 
the  traction  at  the  point  is  given  by  (26). 

17.  The  Magnetic  Field  of  Two  Infinitely  Long  Coaxial  Circular 
Cylindrical  Shells  Traversed  by  the  Same  Current  in  Opposite 
Directions.  Let  /  denote  the  current,  and  FF  and  GG  the 
shells,  Fig.  97. 

The  magnetic  field  in  the  region  A  is  zero,  as  proved  in  a 
similar  case  in  the  last  article. 


306    ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

The  field  in  the  region  C  outside  both  conductors  is  also  zero, 
since  a  closed  curve  drawn  around  the  outer  shell  encloses  equal 
currents  flowing  in  opposite  directions. 

In  the  region  BB  the  field  is  circular,  and  the  intensity  at  the 
distance  d  from  the  axis  is  H  =  IJ2'jrd. 


Fig.  97. 

Within  the  shells  F  and  G  the  field  is  also  circular  in  the  same 
direction,  but  gradually  diminishes  in  intensity  from  B  to  C  and 
from  B  to  A.  These  intensities  are  computed  in  §§  21  and  22, 
XIII. 

The  vanishing  of  the  field  in  the  region  C  might  be  regarded 
as  due  to  the  superposition  of  the  field  connected  with  FF  and 
the  field  connected  with  GG,  the  intensities  of  the  two  being 
equal  and  opposite  at  any  point  of  the  region  CC.  This  would 
also  account  for  the  gradual  diminution  of  the  intensity  in  pass- 
ing through  the  outer  shell  from  B  to  C. 

18.  The  Magnetic  Field  of  Two  Parallel  Circular  Cylindrical 
Conductors  carrying  any  currents  in  the  same  or  opposite  direc- 
tions can  be  obtained  at  once  by  superposing  the  two  fields,  each 
already  obtained  in  §  15.  Maxwell's  diagram  of  the  lines  of 
intensity  when  the  wires  are  linear  is  given  in  Fig.  98  for  the 
case  in  which  the  same  current  traverses  both  wires  in  the  same 
direction  ;  and  the  complete  diagram  is  given  in  Fig.  27,  II.,  for 
the  case  in  which  the  same  current  traverses  the  two  wires  in 
opposite  directions.  When  the  conductors  are  not  linear,  the  same 
diagrams  hold  good  for  the  region  outside  the  conductors,  and  the 
construction  of  the  internal  part  of  the  diagram  offers  no  difficulty. 


CURRENTS    AND    MAGNETIC    FIELDS. 


307 


When  the  currents  are  equal  and  opposite,  the  lines  of  intensity, 
as  shown  in  the  diagram,  are  circles  about  the  wires.      Hence  the 


Fig.  98. 


equipotential  surfaces,  or  the  equipotential  lines  in  the  diagram, 
are  arcs  of  circles  cutting  the  wires.      Compare  §  19,  II. 


Fig.  99. 


That  the  lines  of  intensity  in  this  case  are  circles  may  be  shown 
analytically  as  follows  :  Let  the  plane  of  the  paper  cut  the  wires 
perpendicularly  in  the  points  A  and  B,  Fig.  99,  and  let  PP'  =  ds 


3o8 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


be  the  element  of  a  line  of  intensity.  The  rest  of  the  figure  is 
sufficiently  clear  to  need  no  further  explanation.  Since  there  is 
no  intensity  perpendicular  to  a  line  of  intensity,  we  have  at  PP1  ', 

H^  •  dLJds(  =  sin  0a)  =  H2  •  dLJds(=  sin  0J 

x  •  dL^jds  =  //  2?rZ2  •  dLJds 

.  .   ,  . 

<*A/  A  =  "A/  Ai 


or 

from  which 


Integrating,  we  have 


=  constant 


which  is  the  equation  of  a  circle  with  center  on  AB  produced. 

The  direction  of  the  resultant  intensity  H  is  that  of  the  normal 
CPG  to  the  equipotential  (circular)  arc  APB  through  Pt  Fig.  100. 


It  makes  with  PB  the  angle  90°  +  Ov  since,  as  is  easily  proved 
from  the  figure,  arc  PB  =  arc  FE. 
In  magnitude 


=  /cos  02/27rAP+  /cos  Oj2TrBP 


Exactly  the  same  method  used  in   §  19,  Chapter  II.,   might 
have  been  used  in  the  above  investigation  ;  and  the  method  here 


CURRENTS    AND    MAGNETIC    FIELDS.  309 

used  can  be  applied  with   equal  facility  to  the  problem  of  that 
article. 

19.  The  Magnetic  Field  of  an  Infinite  Solenoid.  Consider  an 
infinitely  long  straight  coil  of  wire  wound  uniformly  and  closely 
at  right  angles  to  its  length,  the  coil  being  of  uniform  cross- 
section  and  shape.  Let  there  be  n  turns  per  unit  length.  Then, 
since  we  have  supposed  the  wires  perpendicular  to  the  axis  of 
the  coil,  the  current  can  be  regarded  as  forming  a  current  sheet 
circulating  as  indicated  in  the  figure  (Fig.  101). 


Fig.  101. 

From  symmetry,  all  the  lines  of  intensity  inside  and  outside 
the  coil  must  be  either  parallel  to  the  axis  of  the  coil  or  in  planes 
perpendicular  to  the  axis  and  continuous  around  it.  By  §  1 1, 
equation  ( 1 2),  the  second  alternative  is  impossible,  hence  all  the 
lines  must  be  parallel  to  the  axis. 

Let  H  and  H'  denote  the  intensities  just  within  and  just  with- 
out the  coil  at  the  point  A,  considered  positive  when  in  the  posi- 
tive direction  through  the  coil  (the  direction  of  the  current  in 
the  solenoid  being  chosen  as  the  positive  direction  around  the 
solenoid)  ;  and  let  the  m.m.f.  be  computed  for  the  path  abcda, 
which  encloses  n  ab  wires  and  therefore  a  current  n  ab  I.  Thus 

m.m.f.  =  H  ab  +  o  be  -f  Hf  cd  -f  o  da  =  (H  —  H'}ab  =  nl  ab 
Hence  H  -  Hf  =  nl 

Since  the  m.m.f.  along  be  and  ad  is  zero,  the  total  m.m.f. 
around  the  circuit  is  independent  of  the  length  of  be  and  ad; 
hence  H  —  H'  is  constant  in  magnitude  as  well  as  in  direction, 
and  is  equal  to  nl. 

If  5  denotes  the  area  of  the  coil,  fiffS  is  the  magnetic  flux 
through  the  coil  and  the  return  flux  outside,  since  the  tubes  of 


310         ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

induction  are  continuous.  Hence  H'  is  opposite  to  H  in  direc- 
tion, and  since  the  flux  returns  parallel  to  the  axis  across  an  in- 
finitely great  area  S',  pH'  =  pHS  S'  y  or  zero.  Hence  H'  =  o, 


and  H=nl  (28) 

everywhere  within  the  solenoid.  The  intensity  in  the  wire  dimin- 
ishes gradually  to  zero  in  passing  from  the  inside  to  the  outside 
of  the  solenoid. 

Since  there  is  no  external  field,  the  inductivity  of  the  external 
medium  may  differ  in  any  way  from  that  of  the  internal  medium 
without  affecting  the  field. 

In  the  case  of  an  actual  solenoid  which  is  long  but  not  infinite, 
the  above  results  are  obviously  approximately  true. 

20.  A.  The  Magnetic  Field  of  a  Finite  or  Infinite  Circular  Sol- 
enoid. The  magnetic  intensity  at  any  point  along  the  axis  of  the 
solenoid,  of  radius  R  and  n  turns  per  unit  length,  can  be  found  by 
direct  integration  from  equation  (16).  From  this  equation  the 
intensity  at  the  given  point  P  due  to  the  current  in  the  infinitesi- 
mal ring  of  width  dx  distant  x  from  P  is 

dH= 


the  current  in  the  ring  being  nldx.     The  total  intensity  at  P  is 
therefore 


fLl< 
J-t, 


(29) 


if  P  is  distant  L^  and  Z2  from  the  ends  of  the  solenoid. 
If  L2  —  Zj  =  L,  this  equation  becomes 

H=nf/(i+R*/Ly  (30) 

(29)  shows  that  if  the  length  of  the  solenoid,  LL  -j-  L2  =  2L,  is 
great  in  comparison  with  R,  the  intensity  along  the  axis,  and 
therefore  (the  magnetic  pressure  ^  pH2  being  remembered)  the 
intensity  throughout  the  volume  of  the  solenoid,  is  very  nearly 
constant  and  equal  to  nl,  except  near  the  ends  of  the  solenoid. 


CURRENTS    AND    MAGNETIC    FIELDS.  311 

In  this  case  the  external  field  of  the  solenoid  is  very  weak 
except  near  the  ends,  and  the  external  medium  may  be  altered 
in  any  manner,  except  in  these  regions,  without  sensibly  affecting 
the  internal  field. 

If  the  flux  through  a  long  slender  solenoid  .is  <f>,  the  magnetic 
field  at  external  points  is  very  nearly  the  same  as  the  field  of  a 
permanent  magnet  placed  coincident  with  the  solenoid  and  having 
two  approximately  concentrated  poles,  of  strength  -f  <J>  and  —  4>, 
at  its  ends. 

When  Z2  =  L^  =  infinity,  (29)  reduces  to  (28). 

B.  A  Very  Long  and  Slender  Cylindrical  Rod  of  Inductivity  /i 
Placed  Within  a  Longer  and  Wider  Uniformly  Wound  Solenoid 
Containing  a  Medium  of  Inductivity  /^  Parallel  to  its  Axis. 
Except  near  the  ends  of  the  rod,  the  demagnetising  intensity 
due  to  its  poles  is  small,  and  if  the  ratio  of  its  length  to  its 
diameter  is  sufficiently  great,  negligible. 

The  magnetic  intensity  within  and  without  the  rod,  except 
near  its  ends  and  the  ends  of  the  solenoid,  is 


The  magnetic  induction  within  the  rod  is 

B=ii, 
and  that  within  the  rest  of  the  core 


the  region  near  the  ends  being  excepted. 

The  intensity   of  magnetisation   of  the  rod,  except  near  its 
ends,  is 

/  =  B  -  B,  =  (fji  -  ^)H 

and  the  numerical  strength  of  each  of  the  poles,  distributed  over 
the  ends  of  the  rod,  is 


m  =  /  5=  (B  -  ^H)S  =  (fji  -  n^HS  (31) 

where  5  denotes  the  area  of  the  cross-section  of  the  rod. 


312 


ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 


21.  The  Magnetic  Field  Between  Two  Infinite  Parallel  Planes 
Traversed  by  Equal  and  Opposite  Currents.  The  result  obtained 
for  the  infinite  solenoid,  (19),  being  independent  of  the  shape  and 
area  of  the  cross-section  of  the  coil,  may  be  applied  to  a  coil 
with  a  rectangular  cross-section  of  finite  height  and  infinite 
width.  We  have  then  two  parallel  plane  current  sheets  in 


Fig.  102. 

opposite  directions.  If  the  current  directions  are  as  shown  in  the 
figure  (Fig.  102),  and  if  the  current  per  unit  length  of  the  coil 
perpendicular  to  the  plane  of  the  figure  is  nl,  the  uniform  mag- 
netic field  of  intensity  H  =  nl  is  directed  perpendicularly  into 
the  plane  of  the  paper.  This  result  could  of  course  have  been 
readily  obtained  independently. 

22.  The  Magnetic  Field  of  an  Endless  Coil  Uniformly  Wound 
Upon  a  Solid  of  Revolution  Generated  by  Revolving  a  Plane 
Area  S  about  an  Axis  in  its  Plane  but  Not  Passing  Through  It 


j 

Fig.  103. 

(Toroidal  Coil).  The  current  is  assumed  to  circulate  in  a  sheet 
accurately  perpendicular  to  the  coil,  as  shown  in  the  figure  (Fig. 
103).  The  cross-section  of  the  coil  is  shown  rectangular  in  the 
figure,  but  may  have  any  shape. 


CURRENTS   AND  MAGNETIC    FIELDS.  313 

By  symmetry,  all  the  lines  of  intensity  must  be  either  circles 
about  the  axis  of  revolution  in  planes  perpendicular  to  the  axis, 
or  closed  curves  around  or  within  the  coil  in  planes  cutting  it 
perpendicularly.  The  second  alternative  is  impossible,  by  §  14, 
since  such  curves  would  enclose  no  current.  For  the  same  rea- 
son there  can  be  no  circular  lines  of  intensity  centered  on  the 
axis  outside  the  coil.  All  such  circles  inside  the  coil,  however, 
enclose  the  total  current  circulating  around  the  coil.  If  the 
total  number  of  turns  of  wire  in  the  coil  is  Nt  and  if  the  current 
strength  is  7,  the  m.m.f.  along  any  circle  of  intensity  of  radius  d 

within  the  coil  is  LT     j       ATT 

H2ird  =  NI 

whence  H  =  NI \2Trd  (32) 

If  the  radius  of  the  shortest  line  of  intensity  is  R,  and  if  there 
are  n  turns  per  unit  length  of  this  line,  N=  2irRn,  and 

H=nIRId  (33) 

If  R  becomes  infinite,  and  if  the  area  of  the  cross-section  of  the 
coil  remains  constant,  so  that  R  and  d  approach  equality  as  they 
approach  infinity,  ,.,__  , 

as  otherwise  shown  in   §  19,  which  with   §  21,  is  thus  a  special 
case  of  the  present  article. 

Since  there  is  no  external  field,  it  is  immaterial  what  medium 
surrounds  the  toroid. 

23.  The  Force  Between  Two  Infinite  Parallel  Linear  Conductors 
Carrying  Electric  Currents.  Let  the  currents,  of  strengths  7L  and 
72,  intersect  the  plane  of  the  paper  at  A  and  B  respectively,  dis- 
tant d  apart,  and  first  suppose  that  the  currents  have  the  same 
direction,  down  into  the  plane  of  the  paper.  Then  the  intensity 
7/2  due  to  the  current  72  is  Ij2ird  at  all  points  of  the  wire  A, 
and  is  directed  vertically  upward  at  right  angles  to  the  wire. 
Hence  the  force  upon  a  length  L  of  the  wire  A  is 


3H          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

t  g^ter  (34) 
the  last  expression  being  the  numerical  value  of  the  force. 

V/j/^  is  directed  toward  Bt  thus  the  force  is  one  of  attraction 
between  the  wires. 

The  direction  of  F  is  not  affected  if  the  directions  of  both 
currents  are  reversed.  If  however  the  direction  of  one  of  the 
currents  only  is  reversed,  F  becomes  a  force  of  repulsion. 

The  fields  surrounding  the  wires  when  the  currents  have  the 
same  magnitude  and  flow  in  the  same  and  opposite  directions  are 
shown  in  Figs.  98  and  27.  The  force  upon  each  wire  is  seen  to 
be  always  from  the  stronger  to  the  weaker  part  of  the  field. 

dX 


24.  The  Forcive  Between  Two  Infinite  Straight  Linear  Conduc- 
tors Perpendicular  to  One  Another  and  distance  d  apart.  Let  one 
of  the  conductors,  AC,  Fig.  104,  lie  in  the  plane  of  the  paper, 
and  let  the  other,  D,  be  perpendicular  to  this  plane,  the  currents, 
/!  and  /2,  being  directed  to  the  right  and  downward  respectively. 
By  considering  the  field  of  one  current  in  the  neighborhood  of 
the  other,  the  forcive  is  seen  to  be  a  pure  torque  tending  to  make 
the  two  currents  flow  in  the  same  direction.  To  obtain  the 
torque  T upon  a  length  L  of  AC,  with  center  at  By  we  have,  for 
the  force  upon  an  element  of  AC  of  length  dx  distant  x  from  B, 

dF  =  BJ^dx  xj(d*  -f  ;r2)*  =  /x//2^/27r(X2  +  *2) 
and  dT  =  xdF  =  pl^dx}  2Tr(dz  +  x*) 

BD  being  chosen  as  axis  of  moments,  and  a  positive  torque  tend- 
ing to  move  C  down.  Hence  the  total  torque  upon  a  length  L 
of  AC  with  center  at  B  is 

r-(L  +  ird-  2d  tan'1  LJ2d)  (35) 


CURRENTS   AND    MAGNETIC    FIELDS.  315 

By  considering  the  field  of  one  wire  in  the  neighborhood  of 
the  other  it  is  easy  to  see  that  when  the  currents  make  with  one 
another  an  angle  6  different  from  90°,  there  is,  in  addition  to  the 
torque  (which  vanishes  when  #  =  o°  or  180°),  a  force  between 
the  wires,  attractive  when  6  is  less  than  90°  and  repulsive  when 
6  is  greater  than  90°,  the  force  upon  each  element  being  always 
from  the  stronger  to  the  weaker  part  of  the  resultant  field. 

25.  The  Force  Between  Two  Linear  Circular  Coaxial  Wires  dis- 
tant d  apart,  where  d  is  very  small  in  comparison  with  the 
radius  of  either  circle.  Since  the  field  very  near  any  thin  wire 
is  approximately  the  same  as  the  field  very  near  an  infinite  straight 
wire,  the  approximate  force  in  this  case  is  easily  obtained. 

If  the  circles  have  the  same  radius,  R,  and  if  the  currents  are 
/!  and  /2,  the  force  is 

/<-  =  //2/2W-  2irR  =  IJJtfd  (36) 

and  is  attractive  or  repulsive  according  as  the  currents  flow  in 
the  same  or  opposite  directions. 

If  the  radii  of  the  coils  are  Rl  and  R2  =  Rl  -f  a,  and  if  the 
distance  between  their  planes  is  b, 

•  '";;;'";'':;;';.    F- */,#/(«•  +  ?)  ;  ;;    :;"",  (37) 

which  is  a  maximum,  for  given  values  of  Rl  and  Rv  when  b  =  a. 

-B&  The  Torque  upon  a  Circular  Cylindrical  Coil  of  n  Turns  and 
Radius  r  Placed  with  its  Center  in  the  Axis  of  Two  Coaxial  Circular 
Coils  each  of  radius  R  and  N  turns,  at  a  point  equidistant  from 
the  planes  of  the  coils,  with  which  its  own  right  cross-section 
makes  an  angle  0,  when  R  is  much  larger  than  r.  The  field  of 
the  larger  coils  will  be  sensibly  uniform  throughout  the  region 
occupied  by  the  smaller  coil  and  equal  to  its  value  at  the  center 
of  the  axis.  If  the  distance  between  the  planes  of  the  two  large 
coils  is  2d,  and  if  the  currents  of  the  larger  and  smaller  coils  are 
/  and  z  respectively,  then,  by  §  13,  if  the  currents  in  the  larger 
coils  have  the  same  direction,  the  field  intensity  at  the  center  of 
the  axis  is 


316          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 


Hence,  by  §  10,  there  is  a  torque  upon  the  smaller  coil  equal  to 

T=  ni^H  cos  (90°  -  0)  •  -rrr2 

2  sin  0/(R2  +  </2)* 


which  is  a  maximum  when  the  smaller  coil  is  perpendicular  to 
the  other  two.  There  is  no  translatory  force  acting  upon  the 
coil  as  a  whole. 

If  the  same  current  /  traverses  both  coils,  and  if  the  medium 
surrounding  the  coils  is  air  or  free  aether  (a  vacuum),  in  which 


T=  P  nNir^R2  sin  0/(R2  +  d^  (39) 

numerically. 

From  the  above  equations,  and  the  equations  developed  in 
foregoing  articles,  for  the  forcive  between  two  conductors  carry- 
ing the  same  current,  the  electromagnetic  unit  current  and  the 
inductivity  of  the  surrounding  medium  can  be  defined  without 
reference  to  magnets  of  any  kind. 

It  can  easily  be  shown  that  the  field  at  the  center  of  the  axis 
is  most  nearly  uniform  when  d  is  made  equal  to  R.  In  this  case 

(39)  becomes 

T=  P  nNirr2  sin  0/2*R  (40) 

numerically. 

27.  Galvanometers.     A  galvanometer  is  an  instrument  for  meas- 
uring or  detecting  electric  currents  by  means  of  the  forcive  acting 
between  a  permanent  magnet  and  a  conductor  traversed  by  an 
electric  current.     There  are  two  general  types  of  such  instru- 
ments :  In  one  the  magnet  is  fixed  and  the  conductor  is  mov- 
able, in  the  other   the   conductor  is  fixed   and   the  magnet   is 
movable. 

28.  The    Deprez-D'Arsonval  Galvanometer.     This  is  an  instru- 
ment of  the  first  type.      It  consists  essentially  of  a  permanent 
horseshoe  magnet  NS,  Fig.  105,  with  a  very  strong  magnetic 
field  between  its  poles  ;  a  coil  C,  consisting  of  many  turns  of  fine 


CURRENTS   AND    MAGNETIC    FIELDS. 


317 


insulated  wire,  suspended  vertically  by  a  very  fine  metallic  ribbon 
AD  .continuous  with  the  wire  of  the  coil ;  and,  attached  to  the 
coil,  a  pointer  or  light  mirror  B,  by  which  any  angular  change 
in  the  position  of  its  plane  may  be  determined. 


Fig.  105. 

The  coil  is  adjusted  until,  when  no  current  is  flowing,  its  plane 
is  parallel  to  the  magnet's  field,  which  is  so  strong  that  the  field 
of  the  earth  and  other  magnets  or  currents  in  the  neighborhood 
is  negligible  in  comparison.  When  traversed  by  a  current,  the 
coil  is  deflected  about  AB  as  axis  until  the  torque  exerted  upon 
it  by  the  magnetic  field  is  balanced  by  the  return  torque  due  to 
the  twist  of  the  elastic  suspension.  Let  6  denote  the  permanent 
angle  of  deflection  when  the  current  strength  is  /,  vS  and  n  the 
(average)  area  of  a  single  turn  of  the  coil  and  the  number  of 
turns,  respectively,  and  B  the  magnetic  induction,  supposed 
uniform,  between  N  and  5.  Then  the  torque  upon  the  coil 
due  to  the  field  is,  by  (9),  §  10, 


If  R  is  the  torsional  constant  of  the  suspending  ribbon,  we 

have  also 

T=Rd 


318          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

Hence,  equating  the  two  values  of  Tand  solving  for  /,  we  have 
f=R/nSB-0/cos0  (41) 

For  small  values  of  6j  cos  0  is  sensibly  equal  to  I,  and  /  is 
sensibly  proportional  to  6. 

By  giving  a  special  shape  to  the  magnet's  pole  faces  and  insert- 
ing within  the  coil  symmetrically  a  cylinder  of  steel  magnetised 
transversely,  or  a  cylinder  of  soft  iron,  the  induction  in  the  region 
moved  through  by  the  vertical  portions  of  the  coil  may  be  made 
practically  uniform  in  magnitude  and  parallel  to  the  plane  of  the 
coil  in  any  position.  In  this  case  T=  ;/5B/=  R6  and 

/=  RjnSB6  (42) 

for  all  angles. 

29.  The  Tangent  Galvanometer.  The  simplest  galvanometer 
of  the  second  type,  in  which  the  magnet  is  movable  and  the 
conductor  fixed,,  is  the  tangent  galvanometer.  This  instrument, 
in  its  simplest  form,  consists  essentially  of  a  circular  coil  of  wire, 
each  turn  having  practically  the  same  radius  R,  wound  upon  a 
suitable  frame  (of  non-magnetic  material)  ;  a  small  permanent 
magnet  suspended  by  a  long  thread  of  quartz  or  silk  as  free  from 
torsion  as  possible,  or  otherwise  mounted  in  such  a  way  as  to 
move  very  freely,  at  the  center  of  the  coil  ;  and  a  mirror  or  a 
pointer  mounted  upon  the  magnet,  by  which  its  angular  motion 
may  be  determined.  The  coil  is  placed  with  its  plane,  or  the 
nearly  coincident  planes  of  its  turns,  in  the  magnetic  merid- 
ian, NS. 

When  there  is  no  current  in  the  coil  the  magnet  will  come  to 
rest  with  its  axis  in  this  plane,  under  the  action  of  the  horizontal 
component  of  the  earth's  magnetic  intensity,  which  will  be  denoted 
by  H.  When  a  current  /traverses  the  coil,  there  is  developed 
at  the  center  of  the  coil  a  magnetic  intensity  perpendicular  to  H 
and  equal  to 


GI 
////=  n/2R)  being  a  constant  for  the  given  coil. 


CURRENTS    AND    MAGNETIC    FIELDS.  319 

The  resultant,  Hf,  of  the  two  fields  makes  with  H  an  angle 


and  the  magnet  will  be  deflected  through  this  angle,  coming  to 
rest  when  its  axis  lies  in  the  vertical  plane  through  Hf  .  Hence 
we  have  /==  2RjnH=  2RUfn-  tan  0  =  H/£-tan  0  (43) 

If  the  magnet  is  suspended  at  the  center  of  the  axis  of  two 
similar  parallel  coils  carrying  the  same  current  in  the  same  direc- 
tion, we  have 

/=  (R*  +  <**)»  H/^2-  tan  0  =  H/C'  -tan  6  (44) 

It  can  easily  be  shown  that  the  field  near  the  center  of  the  axis 
is  as  nearly  uniform  as  possible  when  d  is  made  equal  to  R.  In 
this  case  (44)  becomes, 

7=  2i^H/»-tan0=  H/£"-tan0  (45) 

The  Determination  of  a  Current  in  Absolute  Electromagnetic 
Measure.  By  measuring  all  the  quantities  in  the  second  member 
of  either  of  the  above  equations,  a  current  may  be  determined  in 
terms  of  the  REM  unit. 

30.  Sensitive  Galvanometers  (Second  Type).  Such  a  galvanom- 
eter as  that  just  described,  though  valuable  as  a  means  of  determin- 
ing a  current  in  absolute  measure,  is  by  no  means  sufficiently  sen- 
sitive for  most  purposes  for  which  a  galvanometer  is  needed.  The 
sensitiveness  of  a  galvanometer,  or  the  ratio  of  the  deflection  to 
the  current,  or  change  in  deflection  to  change  in  current,  may 
evidently  be  increased  by  increasing  the  number  of  turns  and 
bringing  them  closer  to  the  magnet,  or  by  diminishing  the 
effect  of  the  external  magnetic  field  acting  upon  the  magnet. 
The  latter  method  will  be  considered  first. 

One  method  of  diminishing  the  strength  of  the  earth's  magnetic 
field  is  by  placing  one  or  more  magnets,  called  control  magnets,  in 
such  positions  as  to  neutralise  to  a  greater  or  less  extent  the  earth's 
field.  This  method,  while  very  generally  practised,  is  open  to  the 
often  serious  objection  that  slight  changes  in  the  earth's  field, 


320          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

which  are  constantly  occurring,  produce  very  large  percentage 
variations  in  the  weak  resultant  field  at  the  magnet,  thus  mak- 
ing the  behavior  of  the  instrument  extremely  inconstant. 

A  better  method  is  to  surround  the  galvanometer  with  a  thick- 
walled  case  of  iron,  or  several  such  cases,  to  act  as  a  magnetic 
shield.  This  greatly  reduces  the  field  of  the  earth  without  in- 
creasing the  relative  prominence  of  its  variations  (§23,  XL;  §§ 
14-1 5,  IV.). 

A  still  better  method,  now  always  adopted  for  the  most  sen- 
sitive instruments,  usually  in  conjunction  with  at  least  one  of  the 
other  two  already  mentioned,  is  to  mount  on  the  same  suspen- 
sion two  galvanometer  magnets  of  as  nearly  as  possible  the  same 
moment,  with  their  poles  turned  in  opposite  directions.  If  the 
axes  of  these  magnets  lie  accurately  in  the  same  plane,  and  if 
their  moments  about  the  axis  of  suspension  are  rigorously  equal, 
the  earth's  field  can  exercise  no  directive  influence  upon  the  sys- 
tem. Such  a  magnetic  system  is  said  to  be  astatic.  With  this 
condition  approximately  realised  in  practice,  the  directive  effect 
of  the  earth's  field  can  be  made  extremely  small.  The  current 
is  passed  around  one  magnet  only,  or  else,  in  highly  sensitive 
instruments,  in  opposite  directions  around  the  two  magnets.  A 
control  magnet  is  used  to  adjust  the  position  of  the  mirror  as 
well  as  to  regulate  the  sensitiveness  of  the  instrument. 

31.  The  Winding-  of  a  Sensitive  Galvanometer  of  the  Second 
Type.  With  respect  to  the  winding  of  a  sensitive  galvanometer 
of  the  second  type,  it  is  evident  that  the  magnetic  intensity  at  a 
point  0,  the  position  of  the  magnet,  Fig.  106,  due  to  unit  current 
in  length  L  of  wire  wound  upon  a  circular  arc  whose  radius  sub- 
tends an  angle  6  at  O  and  every  point  of  which  is  distant  r  from 

0,  is 

H=  L  sin  6/47rr2  (46) 

If  the  same  wire  is  wound  in  parallel  circles  with  their  axes 
through  0  anywhere  on  the  surface  of  revolution  the  equation  of 
whose  generating  curve  is  sinO/r2  =  47T///Z,  =  constant,  the 


CURRENTS    AND    MAGNETIC    FIELDS. 


32I 


field  intensity,  H,  at  0  will  be  the  same  as  that  given  by  (46). 
This  equation  may  be  written 

sin  0/r2  =  i  //  or  r2  =  p2  sin  d  (47) 

where  p*  =  constant  =  Lj^rrH. 

A  plane  section  through  the  axis  of  revolution  of  this  surface, 
drawn  for  a  given  value  of  /2,  is  shown  in  Fig.  106. 

Since  all  points  within  a  surface  drawn  for  a  given  value  of 
/  lie  upon  surfaces  with  smaller  values  of/,  and  therefore  greater 
values  of  H\  and  since  all  points  without  the  surface  lie  upon 
surfaces  with  greater  values  of  p  and  smaller  values  of  H ;  it 
follows  that  a  given  length  of  wire,  in  order  that  it  may  produce 


Fig.  106. 

the  maximum  field  at  0,  should  be  wound  so  that  its  whole  mass 
just,  fills  up  the  volume  within  the  surface  given  by  the  equation 
(47),  where  the  magnitude  of  /  depends  upon  the  quantity  and 
the  length  of  wire.  In  the  actual  winding,  of  course,  some  space, 
indicated  in  the  figure  by  dotted  lines,  must  be  left  for  the  magnet 
and  suspension.  These,  however,  in  modern  sensitive  galva- 
nometers, are  so  small  as  to  weigh  but  a  very  small  fraction  of 
a  gram,  so  that  this  space  is  not  large. 

Since  a  given  length  of  fine  wire  occupies  less  volume  than 
the  same  length  of  coarser  wire,  it  is  an  advantage,  if  a  galva- 
nometer of  a  certain  resistance  is  to  be  constructed,  to  use  fine 


322          ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

wire  for  the  smaller  values  of  /,  that  is,  near  the  magnet,  and 
coarser  wire  further  from  the  magnet.  Three  sizes  of  wire  are 
frequently  so  used  in  the  same  coil,  although  a  single  wire  of 
uniform  cross-section  is  usually  employed. 

It  is  evident  that  the  current  traversing  the  coils  of  a  galva- 
nometer constructed  as  above  is  not,  in  general,  strictly  pro- 
portional to  the  tangent  of  the  angle  of  deflection.  The  law 
/=  \r\jG'F(6),  where  £=////,  connecting  the  current  with 
the  deflection  can,  however,  easily  be  determined  by  experiment. 

32,  The  Ballistic  Galvanometer.  The  electric  charge  of  a  con- 
denser, or  the  charge  circulating  through  a  coil  owing  to  electro- 
magnetic induction  (XIII. ),  that  is,  in  general,  the  time  integral 
of  a  transitory  current,  can  be  measured  with  a  galvanometer, 
provided  that  sensibly  the  whole  charge  can  be  made  to  circulate 
through  the  galvanometer's  coil  before  the  magnet  (or  coil,  if 
the  instrument  is  of  the  first  type)  has  moved  appreciably*  from 
its  position  of  equilibrium,  and  provided  that  the  damping  (or 
retardation  due  to  friction,  induced  currents  (XIII.,  §  5),  etc.)  of 
the  magnet's  (or  coil's)  motion  is  but  slight.  To  insure  that 
these  conditions  shall  be  satisfied,  and  to  make  the  deflections  of 
the  mirror  easy  to  read,  a  galvanometer  designed  for  this  purpose, 
called  a  ballistic  galvanometer,  is  constructed  with  a  magnet  (or 
coil)  of  considerable  moment  of  inertia  (and  long  period),  and 
damping  is  prevented  as  far  as  possible. 

We  shall  consider  here  only  an  instrument  of  the  second  type. 
Let  H  denote  the  magnetic  intensity  at  the  magnet  due  to  the 
earth  and  the  control  magnets,  M  the  moment  of  the  magnet, 
K  its  moment  of  inertia,  H  the  magnetic  intensity  at  the  magnet 
due  to  a  current  7  in  the  galvanometer  coils,  and  G  the  constant 
ratio  of  H  to  /,  as  in  §  29.  Then 

H=  GI  =  G  dgjdt 
when  a  current  /=  dqjdt  traverses  the  coils. 

*The  cosine  of  the  angle  moved  through  by  the  magnet  in  the  discharge  lime  t? 
must  not  differ  sensibly  from  unity,  or  a  correction  must  be  applied. 


CURRENTS    AND    MAGNETIC    FIELDS.  323 

The  torque  upon  the  magnet,  which  remains  sensibly  in  its 
position  of  equilibrium  during  the  passage  of  the  whole  (appre- 
ciable part  of  the)  charge,  due  to  the  current  /  is 


=  GMdqjdt 

When  a  total  charge  q  is  passed  through  the  coils,  the  (appre- 
ciable part  of  the)  transitory  current  lasting  for  a  time  t'  so  small 
that  the  magnet  does  not  sensibly  move  within  this  time,  the 
total  angular  impulse  upon  the  magnet  is 

f  Tdt  =  GM\    (dqjdt}dt  =  GM  ^  dq  =  GMq 

t/O  *J$  t/ 

If  a)  denotes  the  angular  velocity  imparted  to  the  magnet  by 
the  discharge 


I 

Jo 


Tdt  =  GMq  =  K 


and  the  kinetic  energy  of  the  magnet  just  after  the  discharge, 
while  still  sensibly  in  its  position  of  equilibrium,  is 


If,  as  we  assume  for  the  present,  there  is  no  damping,  the 
magnet  will  come  to  rest  at  a  certain  angle  of  deflection  6,  such 
that  the  work  done  against  the  magnetic  field  of  intensity  H  is 
equal  to  the  original  kinetic  energy  \KaP.  (The  torsion  of  the 
fiber  suspending  the  magnet  is  supposed  negligible.)  Hence 

\(GMqJIK=    PVlH  sin  OdO  =  MH(i  -  cos  0) 

=  2MH  sin2J0 
from  which 

sin  \G  (48) 


Thus  the  charge  is  proportional  to  the  sine  of  one-half  the  angle 
of  deflection  (first  elongation). 

To  obtain  the  charge  in  absolute  measure  from  (48),  it  would 
be  necessary  to  know  both  KandM,  or  their  ratio,  as  well  as  H 


324          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

and  G  ;  but  the  first  two  of  these  quantities  can  be  eliminated  by 
making  use  of  (34),  XL,  according  to  which 

(KjM)l  =   TV  H/27T 

T  being  the  time  in  which  the  magnet,  vibrating  through  an  in- 

finitesimal arc,  executes  a  complete  oscillation.      Thus  we  have 

q=  HT/irG-sm  \6  (49) 

H/£  can  be  readily  determined  by  passing  a  known  steady 
current  /  through  the  coil  and  noting  the  steady  deflection  0. 
Then,  with  the  nomenclature  of  §  33,  (i), 


The  period  T  of  a  complete  vibration  of  the  magnet  for  an 
infinitesimal  arc  can  be  computed  without  difficulty  from  the 
observed  time  of  vibration  for  a  finite  arc.  For  small  arcs  the 
difference  between  the  two  times  of  vibration  is  not  appreciable. 
If  the  damping  and  the  torsion  of  the  suspension  are  not  negli- 
gible, further  corrections  must  be  applied.  All  these  matters 
are  discussed  in  Gray's  Absolute  Measurements  in  Electricity  and 
Magnetism. 

The  development  of  the  formula  corresponding  to  (49)  for  the 
ballistic  Deprez-D'Arsonval  galvanometer  is  not  difficult  and  is 
left  to  the  reader.  It  is 

q  =  i/KR/nSB  •  6  =  RjnS  B  •  T/TT  .  10  (50) 

The  constant  R/nSB  can  be  easily  determined  from  (41)  by 
means  of  a  known  steady  current  and  the  corresponding  known 
deflection. 

In  what  follows  (49)  is  adopted  as  a  standard  formula  for  the 
ballistic  galvanometer.  If  an  instrument  of  the  first  type,  which 
offers  important  advantages  in  many  cases,  is  used,  the  equa- 
tions must  be  modified  by  the  substitution  of  (50)  for  (49). 

33.  The  Best  Resistance  for  a  Galvanometer  Wound  with  Uni- 
form Wire.  We  shall  now  determine  the  best  resistance  to  be 
given  to  a  set  of  galvanometer  coils,  or  a  galvanometer  coil,  of 
given  type,  in  order  to  produce  the  maximum  sensitiveness, 
when  in  addition  to  the  type  of  galvanometer,  the  space  to  be 


CURRENTS   AND    MAGNETIC    FIELDS.  325 

filled  by  the  wire,  its  disposition  as  to  the  magnet,  and  its  specific 
resistance  are  given.  The  volume  occupied  by  the  insulation 
will  be  supposed  negligible.* 

Let  r  denote  the  volume  to  be  occupied  by  the  wire,  and  L  the 
length,  5  the  cross-section,  r  the  specific  resistance,  and  g  the 
total  resistance,  of  the  wire.  Then 

g-rL\S-rDlT  (a) 

(i)  First  consider  a  galvanometer  designed  to  measure  steady 
currents.  The  current  /  in  the  galvanometer  is  proportional  to 
some  function  of  0,  the  angle  of  deflection,  as  tan  0,  sin  0, 
6  1  'cos  0,  etc.  Let  this  function  be  denoted  by  F(0).  Then  we 
have,  by  what  has  just  been  said, 

F(6)  =  BI  (b) 

in  which  B  is  a  constant  for  the  given  galvanometer  and  coil.  B 
is  evidently  proportional  to  the  length  of  the  wire  in  the  coil. 
That  is 

B  =  KL 

where  K  is  a  constant  depending  on  the  size  of  the  coils,  the 
type  of  instrument,  etc.  (a)  may  therefore  be  written 

F(&)  =  BI=  KLI 
or 


If  the  galvanometer  is  connected  in  circuit  with  a  generator 
with  an  e.m.f.  M*  and  a  resistance  such  that  the  total  resistance 
in  the  circuit  outside  the  galvanometer  is  R,  we  have 


or 

F(&)  =  K-VL\(g  +  R)  =  KVL/(rL2/T  +  R) 

*  For  a  more  complete  discussion  reference  must  be  made  to  Gray's  Absolute 
Measurements  in  Electricity  and  Magnetism  ,  Vol.  II.,  and  to  The  Physical  Review, 
Vol.  V.,  p.  300. 


326          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

So  far  as  its  dependence  upon  the  length  of  the  wire  is  con- 
cerned, F(&)  will  be  a  maximum  when  dF\6)jdL  =  o,  that  is  when 


dF(6)jdL  =  0=  K^(R  -  rL2/r)/(R  +  rZ2/r)2  =  o 
which  gives  rL2/r  =  g  —  R  (51) 

So  that  the  greatest  sensitiveness  will  be  attained  by  giving 
the  wire  such  a  length,  or  such  a  cross-section,  that  its  resistance 
is  equal  to  the  external  resistance.  It  is  obvious  that  r  should 
be  as  small  as  possible  to  produce  the  maximum  current  with  a 
given  e.m.f.  Hence  the  coils  are  usually  wound  of  copper  wire. 

(2)  If  the  galvanometer  is  one  designed  for  measuring  con- 
denser charges  so  that  the  total   charge  q  crosses  every  section 
of  the  wire,  whatever  its  length  or  resistance,  we  have  (§  32) 

Ft(ff)  =  CBq  =  CKLq  (52) 

where  C  is  a  constant  and  B  —  KL  can  be  determined  by  measur- 
ing F(0),  §  33,  (i),  when  a  steady  current  /is  passed  through  the 
coils. 

From  (52)  it  is  clear  that  the  wire  should  be  as  fine  as  possible, 
or,  for  a  given  kind  of  wire,  the  resistance  as  great  as  possible. 
The  specific  resistance  is  immaterial,  provided  that  the  total  re- 
sistance is  not  so  great  as  to  make  the  time  constant  (§41,  Chap- 
ter XIII.)  noticeable. 

(3)  If  the  galvanometer  is  to  be  used  for  measuring  discharges 
produced  by  changing  the  magnetic  flux  through  a  coil  (§9, 
Chapter  XIII.),  we  have, 

F2(6)  =  CBq  =  CKLN\(g  +  R) 

or,  for  a  given  value  of  N  (the  change  of  coil  flux  producing  the 
discharge), 

F2(0)  =  constant  *L/(g  +  R)  =  constant  xL/(R  +  rL2/r) 
Hence  dF2(0)/dL  =  o,  and  the  sensitiveness  is  a  maximum, 
when  g-R  (53) 


CURRENTS   AND    MAGNETIC    FIELDS.  327 

as  in  an  instrument  used  for  steady  currents.      In  this  case  also, 
r  should  be  as  small  as  possible. 

34.  The  Electrodynamometer.  The  formulae  developed  in  §  26 
are  utilised  for  the  absolute  determination  of  current  strength  by 
the  electrodynamometer. 

The  Electrodynamometer  of  Weber,  in  its  simplest  form,  con- 
sists essentially  of  two  large  coils  and  a  much  smaller  coil, 
similar  to  those  described  in  §  26,  together  with  a  fine  metallic 
ribbon  joined  to  one  end  of  the  smaller  coil  and  suspending  this 
coil  from  a  fixed  torsion  head,  with  its  center  at  the  center  of 
the  axis  of  the  two  large  coils,  a  straight  vertical  piece  of  wire 
joined  to  the  other  end  of  the  small  coil  and  dipping  into  a  cup 
of  mercury  below,  and  a  light  mirror  mounted  upon  the  small 
coil  for  reading  its  deflections. 

The  suspension  is  adjusted  by  turning  the  torsion  head  until, 
when  there  is  no  current,  the  planes  of  the  smaller  and  larger 
coils  are  perpendicular.  When  the  same  current  /  is  passed 
through  all  the  coils  in  series,  flowing  in  the  same  direction 
through  the  two  larger  coils,  the  smaller  coil  will  be  deflected 
through  an  angle  6  such  that  the  return  torque  due  to  the  torsion 
of  the  suspending  ribbon  just  balances  the  torque  of  the  field. 
If  Kis  the  constant  of  torsion  of  the  ribbon,  this  torque  is  T— 
KO.  Hence,  by  §  26, 

cos 


and  /  =  K\R*  +  dj  /  '  rR(nN^(0  /  'cos  0)1  (54) 

If  d  =  R,  when  the  field  throughout  the  smaller  coil  is  most 
nearly  uniform, 

/=  2i(RK)l/r(nNfJL'7r)\0/cos  (9)4  (55) 

fi  being  sensibly  equal  to  I  ,  numerically,  when  the  coils  are  in  air. 

The  Siemens  Electrodynamometer.  Instead  of  keeping  the 
upper  end  of  the  suspension  fixed,  and  measuring  the  angle  of 
twist  of  the  lower  end,  the  torsion  head  is  often  turned  in  the  di- 


328          ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

rection  opposite  to  that  of  the  deflection  until  the  deflection  is 
reduced  to  zero.  If  6  is  the  angle  through  which  the  torsion 
head  is  turned  to  effect  this  result,  we  have,  by  §  26,  since  the 
planes  of  the  movable  and  fixed  coils  remain  perpendicular, 


T=K6  = 
and 

/  =  (j?  +  d^(KO^I(nN^rR  (56) 

If  d  =  R, 

/=  2*(KR)Wlr(nNiirn)*  (57) 

Since  all  the  quantities  in  the  second  members  of  (54)  and  (56) 
can  be  determined  by  direct  measurement,  they  suffice  to  deter- 
mine /  in  absolute  measure.  Since  T  is  proportional  to  the 
square  of  /,  an  alternating  current  can  be  measured  with  the 
electrodynamometer. 

Instead  of  a  suspension  made  of  a  single  ribbon,  a  bifilar  sus- 
pension is  often  used  in  the  Weber  electrodynamometer.  It  con- 
sists of  two  fine  wires,  parallel,  or  nearly  parallel,  connected  to 
the  two  ends  of  the  coil,  serving  to  carry  the  current  to  and  from 
the  coil.  For  small  angles  the  formulae  are  the  same  for  the 
two  kinds  of  suspension. 

When  an  electrodynamometer  is  not  to  be  used  for  the  abso- 
lute determination  of  current  strength,  but  only  for  relative 
measurements,  or  when  it  is  to  be  calibrated  by  comparison  with 
standard  instruments,  its  construction  is  often  greatly  modified  to 
increase  or  diminish  the  sensitiveness,  to  reduce  the  size,  etc. 

In  an  electrodynamometer  of  the  Siemens  type,  in  which  the 
movable  coil  is  always  in  the  same  position  when  measurements 
are  made,  the  current  is  always  proportional  to  the  square  root 
of  the  angle  of  torsion,  whatever  the  form  of  the  coils. 

35.  The  Comparison  of  E.M.F.s  by  Poggendorff's  Method,  Ray- 
leigh's  Modification.  A  constant  current  /  is  passed  through  the 
circuit  ABCDA,  Fig.  107,  containing  two  accurately  adjustable 
resistances  R  and  R'  ,  by  a  battery  D  of  constant  e.m.f.  VP0 
greater  than  any  of  the  e.m.f.s  to  be  compared.  The  agents  I'\ 


CURRENTS  AND  MAGNETIC  FIELDS.  329 

and  F2  whose  e.m.f.s  "^1  and  W2  are  to  be  compared  are  placed, 
one  at  a  time,  in  a  shunt  circuit  A  GB,  containing  a  galvanometer 
G  and  a  key  KA,  in  such  a  direction  as  to  oppose  a  current 
through  the  shunt  due  to  D.  The  resistances  R  and  Rf  are  then 
adjusted,  while  their  sum  R  +  Rf  is  kept  constant  and  high,  until 
the  galvanometer  needle  remains  undeflected  whether  the  key 


Fig.  107. 

KA  is  open  or  closed.  Then  ¥",  the  e.m.f.  of  the  agent  Ft  is 
equal  to  RI,  the  potential  difference  between  A  and  B  due  to  the 
field  of  the  current.  Hence,  if  Rl  denotes  the  value  of  R  for  the 
balance  when  F^  is  in  the  circuit,  and  Rz  the  corresponding  value 
when  F2  is  in  the  circuit, 


and  ^i/*,  =  *i/*,  (58) 

Slight  modifications  of  this  method  (potentiometer  methods) 
enable  comparisons  to  be  made  between  two  resistances  or  two 
currents,  and  are  extensively  used. 

36.  The  Comparison  of  a  Capacity  and  a  Resistance  by  the 
Method  of  Direct  Deflections.  A  constant  current  /  traverses  a 
circuit  of  high  resistance  including  in  series  the  resistance  R  under 
experiment.  The  plates  of  the  condenser,  of  capacity  S,  are  con- 
nected to  the  terminals  of  R,  thus  coming  to  a  difference  of  poten- 
tial RI  and  acquiring  a  charge  SRI.  The  condenser  is  then 
insulated  from  the  battery  circuit  and  discharged  through  a 
ballistic  galvanometer  G,  producing  an  angular  deflection  0  such 
that 

SRI=  HTyCV-sinitf  (a) 


330         ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

The  galvanometer  is  then  disconnected  from  the  condenser  and 
its  terminals  connected  to  two  points  of  the  battery  circuit  be- 
tween which  the  resistance  is  r,  a  very  small  fraction  of  the  re- 
sistance W(  which  may  be  made  as  large  as  desired)  from  terminal 
to  terminal  through  the  galvanometer.  This  will  not  alter  the 
current  /  appreciably,  but  will  produce  in  the  galvanometer  a 
constant  current 

=  \\IG-  F(6')  (b) 


6f  being  the  corresponding  steady  deflection.      Eliminating  H/6Y 
from  (a)  by  means  of  (b\  we  obtain 


SR  =  \rj(r  +  W)-}  T/TT  -  sin  ±0  J  F(0>)  (59) 

The  ratio  r/(r  +  W\  Ty  0,  and  F(0f)  being  observed,  SR  is 
given  in  absolute  measure  by  (59). 

The  Determination  of  a  Resistance  in  Absolute  Measure.  Since 
the  capacity  of  a  condenser  can  be  calculated  from  its  dimen- 
sions, the  method  affords  an  absolute  determination  of  a 
resistance. 

The  Comparison  of  Capacities,  E.M.F.s,  etc.,  by  Direct  Deflec- 
tions. The  above  disposition  of  apparatus,  slightly  modified, 
enables  a  comparison  to  be  made  between  two  capacities,  two 
e.m.f.s,  two  resistances,  or  two  currents. 

Thus,  if  two  condensers  with  capacities  Sl  and  S2  are  charged 
to  the  same  voltage  (as  RI)  and  then  discharged  separately 
through  the  same  ballistic  galvanometer,  producing  angular 
throws  of  the  needle  equal  to  6l  and  02,  we  have,  from  (a), 

SJS2  =  sin  J^/sin  J02  (60) 

Or,  if  the  same  condenser  is  charged  to  different  voltages  V^ 
and  V2  in  succession,  and  discharged  each  time  through  the  same 
ballistic  galvanometer, 

sinJ^/sm^  (61) 


CURRENTS    AND    MAGNETIC    FIELDS. 


331 


37.  The  Bridge  Method  of  Comparing  Capacities.  The  conden- 
sers whose  capacities  Sl  and  S2  are  to  be  compared  are  arranged 
as  shown  in  Fig.  107,  Rj_  and  R2  being  adjustable  resistances.  K^ 
is  first  closed,  K2  being  open.  The  condensers  are  now  charged 
to  voltages  Vl  and  Vv  such  that  St  V^  =  S2  V2,  since  they  are 
connected  in  series.  K^  being  kept  closed,  K2  is  then  closed 
•also.  If  Vl  =  ./?!/  and  V2  =  Rzf,  where  /  is  the  steady  current 


Si 

S2 

wvw 

V 

VvAA/WW 

V 

WSAA/ 

{-2  ?• 

x^                     1  

Fig.  108. 

through  Rl  and  R2,  the  needle  of  G  will  remain  undeflected. 
Otherwise  it  will  be  deflected.  In  this  case  the  condensers  are 
discharged  by  opening  K^  K2  being  still  closed,  and  the  process 
is  repeated  with  different  ratios  of  R^  to  R2  until  there  is  no  de- 
flection on  closing  K2  after  K^  (or  on  opening  K^  before  K^. 

Then 


whence  SJS2  =  R^R^  (62) 

The  battery  and  galvanometer  may  be  interchanged  in  the 
above  arrangement.  The  ratio  of  the  capacities  is  given  by  the 
same  formula,  (62). 


CHAPTER    XIII. 

ELECTROMAGNETIC    INDUCTION. 

1.  Magnetic  Flux  Through  a  Coil,     Consider  a  thin  conductor 
in  the  form  of  a  coil  of  n  approximately  closed  turns,  I,  2,  ••-,?/. 
Let  the  positive   direction  around  any  one  of  the  turns,  k,  be 
chosen  arbitrarily,  and  let  the  positive  direction  around  each  of 
the  others  be  chosen  to  coincide  with  the  direction  of  its  current 
when  the  current  through  the  coil  traverses  turn  k  in  the  positive 
direction.      Let  the  magnetic  flux  through  each  turn  be  denoted 
by  <£>  with  the  appropriate  subscript,  as  Qv  <I>2,  .  •  • ,  or  <3>n,  and 
let  <3>  denote  the  average  flux  through  a  single  turn.     Then  the 
^magnetic  flux  through  the  coil,  or  the  coil  flux,  which  will  be  de- 
noted by  N,  is  defined  by  the  equation 

N=  (^  +  <£2  +  .  .  .  +  4>n  =  ;?<£  (l) 

If  the  area  of  the  surface  with  its  edge  in  any  turn  is  not  so 
great  in  comparison  with  the  width  of  the  conductor  that  the 
flux  across  the  conductor  itself  can  be  neglected  in  comparison 
with  the  flux  through  the  turn,  an  approximately  correct  result 
can  be  obtained  by  assuming  the  conductor  (supposed  circular) 
replaced  by  a  linear  conductor  coinciding  with  its  axis. 

2.  The  Inductance,  or  Coefficient  of  Self  Induction,  of  a  coil  or 
circuit  is  defined  to  be  the  quotient  of  the  coil  flux,  Nt  due  to  the 
coil's  own  magnetic  field  divided  by  the  current  /  in  the  coil, 
and  will  be  denoted  by  L.     Expressed  in  the  form  of  an  equation, 
this  relation  is  L=N/I  (2) 

If  the  inductivity  /*  is  constant  throughout  the  magnetic  field  (in- 
dependent of  /)  so  that  B  and  N  are  proportional  to  the  current 

332 


ELECTROMAGNETIC    INDUCTION.  333 

/,  the  inductance  is  the  coil  flux  per  unit  current,  and  is  constant. 
If  the  coil  contains  iron,  L  is  far  from  constant. 

3.  Mutual    Inductance   or    Coefficient    of  Mutual   Induction. 
The  coefficient  of  induction  of  a  coil  I  with  respect  to  a  coil  2  is 
defined  as  the  quotient  of  the  coil  flux  7V12  through  coil  2  due  to 
the  magnetic  field  of  coil   I  divided  by  the  current  7X  of  coil  I, 
and  will  be  denoted  by  J/12.     Thus 

Mv-NJI^    *1o,fi   :      :.    ,T»O-       (3) 

which  is  constant,  for  a  fixed  configuration  of  the  conductors, 
when  /JL  is  constant. 

It  will  be  shown  later  (§13)  that  when  p  is  constant  the  coeffi- 
cient of  induction  of  coil  2  with  respect  to  coil  I  is  equal  to  that 
of  coil  i  with  respect  to  coil  2.  Hence  we  may  write 

5  ;;!-,  Jfa  =  M*  =  M  =  NJI,  =  Na/ft  (4) 

This  relation  is  true  only  when  /4  is  constant.  M=  M12  =  M2l 
is  called  the  coefficient  of  mutual  induction,  or  the  mutual  induct- 
ance, of  the  two  coils  I  and  2. 

The  coefficients  of  self  and  mutual  induction  will  be  defined 
from  energy  considerations  in  §  17. 

4.  Electromagnetic  Induction.     Motional  Electric  Intensity.     It 
follows  as  a  generalisation  from  experiment  that  whenever  a  con- 
ductor moves  in  a  medium  (aether  or  aether  permeated  by  mat- 
ter) supporting  a  magnetic  field  there  is  developed  at  every  point 
P  of  the  conductor  an  intrinsic  electric  intensity  (arising  from  the 
transformation  of  mechanical  energy  into  electrical  or  electrical 
into  mechanical)  equal  to 

e  =  MuB  sin  7  (5) 

where  B  denotes  the  magnetic  induction  at  P,  u  the  velocity  of 
the  point  P  of  the  conductor  with  respect  to  the  medium  (and 
the  fixed  magnetic  induction  at  P),  and  7  the  angle  between  the 
directions  of  u  and  B, 

It  is  often  convenient  to  think  of  the  conductor  as  fixed  and 
the  medium  supporting  the  magnetic  field  as  moving,  together 


334          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

with  the  field,  in  the  opposite  direction  with  the  velocity  u  at  the 
point  P.     In  this  case  we  must  replace  (5)  by 

e  =  NBu  sin  7  (50) 

This  will  be  done  in  what  follows. 

The  electric  intensity  e  is  called  a  motional  electric  intensity. 
It  is  also  very  generally  called  an  induced  intensity. 

A  motional  electric  intensity  in  insulators  has  not  yet  been 
observed.  See  Blondlot,  Journal  de  Physique,  Jan.,  1902. 

Motional  E.M.F.  It  follows  from  (50)  that  the  component  of 
e  at  the  point  P  in  any  direction  is  equal  to  the  vector  product 


Fig.  109. 

of  the  component  of  B  perpendicular  to  this  direction  and  the 
component  of  u  perpendicular  to  the  plane  containing  B  and  the 
given  direction. 

Hence  it  follows  also  that  whenever  an  element,  of  length  dL, 
of  a  line  in  any  conductor  is  moving  in  a  medium  supporting  a 
magnetic  field,  a  motional  electromotive  force  is  developed  along 
dL  equal  to  dL  multiplied  by  the  vector  product  of  the  com- 
ponent of  B  perpendicular  to  dL  and  the  component  of  u,  the 
velocity  of  the  medium  (and  tubes  of  induction)  relative  to  dL 
considered  as  fixed,  perpendicular  to  the  plane  containing  B  and 


ELECTROMAGNETIC    INDUCTION.  335 

dL.  Thus,  if  0  denotes  the  angle  between  B  and  the  component 
of  e  parallel  to  dL,  and  ft  the  angle  between  u  and  the  plane 
containing  B  and  the  component  of  e  just  mentioned  (or  the  ele- 
ment dL),  the  e.m.f.  along  dL  is  given  in  magnitude  and  direction 

by  the  equation      .,        jr\//r>    •     a\/      •     o\  /*\ 

aw  =  dLM(B  sin  &)(u  sin  p)  (6) 

The  relative  directions  of  the  quantities  occurring  in  this  equation 
are  shown  in  Fig.  1 09,  the  axis  of  Z  being  made  to  coincide  with 
the  direction  of  dL  and  dW,  and  the  plane  of  XZ  with  the  plane 
containing  B  and  dL. 

The  second  member  of  (6)  is  evidently  the  time  rate  at  which 
magnetic  flux  sweeps  across  the  element  of  length  dL,  or,  nu- 
merically, the  number  of  unit  tubes  of  induction  moving  across 
dL  per  unit  time.  If  we  denote  this  rate,  per  unit  length  of  dL, 
by  dfyjdt  (6)  may  be  written 

d^=*dL-d$ldt  (6a) 

which,  however,  does  not  give  the  direction  of  dW  (see  Lens's 
law  below). 

From  the  two  preceding  equations  it  is  clear  that  the  motional 
electromotive  force  along  a  line  L  of  any  length  or  form  in  a 
conductor  moving  in  a  medium  supporting  a  magnetic  field  is 

¥  =  fdLV(B  sin  0)(u  sin  /3)  =  fdL  •  dfyjdt  =  d&fdt     (7) 

where  dQjdt  denotes  the  time  rate  at  which  magnetic  flux  sweeps 
in  the  same  general  direction  across  the  line  L,  or,  numerically, 
the  number  of  unit  tubes  of  magnetic  induction  moving  across  L 
per  unit  time,  the  last  two  expressions  not  giving  the  direction 
of  >Jr.  (See  Lenz's  law  below.) 

Motional  e.m.f.  and  Ampere's  Law.  The  existence  of  a  mo- 
tional e.m.f.  and  its  origin  in  the  transformation  of  mechanical 
work  (in  moving  the  conductor  against  the  electromagnetic  forces 
of  §  4,  XII.)  into  electrical  energy  or  vice  versa  being  established 
by  experiment,  its  magnitude  and  direction  can  be  deduced  from 
Ampere's  law  if  we  assume  that  all  the  work  done  in  moving  a 


336         ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

conductor  against  electromagnetic  forces  is  transformed  into 
electrical  energy,  and  that  all  the  energy  acquired  by  a  moving 
conductor  owing  to  the  action  of  electromagnetic  forces  is  trans- 
formed without  loss  from  the  energy  of  the  electromagnetic  field. 
Thus  suppose  dL  at  the  point  P  to  be  the  element  of  length 
of  a  thin  wire  carrying  a  current  /  (due  wholly  or  in  part  to  the 
motional  e.m.f.).  The  force  upon  the  element,  as  given  by 
Ampere's  law,  §  4,  XII.,  is 

dF=dLVIB  sm6 

The  component  of  dF  in  the  direction  of  the  velocity  u  is  then 
dFr  =  dF  sin  ft  =  dL  sin  0  IB  sin  6 

and  if  the   element  is  moved   against  the  force  dF1  with  the 
velocity  —  u  work  will  be  done  upon  the  element  at  the  rate 

P=  dF'u  =  dL  u  sin  $  IB  sin  0 

Hence  the  intrinsic   e.m.f.    developed  in  the  length  dL  by  the 
relative  motion  of  conductor  and  field  is 


=  P//=  dL  u  sin  /3  B  sin  6 
in  magnitude,  and 

dV  =  dLV(B  sin  0)(u  sin  0) 

in  both  magnitude  and  direction,  since  d'W  must  have  the  same 
direction  as  /  when  energy  is  transformed  from  mechanical  to 
electrical  form. 

The  agreement  between  this  result  and  (6)  justifies  the  above 
assumption. 

5.  Induced  Electric  Intensity  and  E.M.F.  It  follows  from  ex- 
periment that  in  a  conductor  at  rest  in  the  surrounding  medium 
the  same  electric  intensities  (and  e.m.f.s)  are  developed  by  a 
given  motion  of  the  tubes  of  magnetic  induction  (without  motion 
of  the  medium)  relatively  to  the  conductor,  due  to  the  motion  in 
the  medium  of  the  magnets  or  electric  circuits  producing  the  field, 
as  by  the  same  relative  motion  of  the  conductor  with  reference  to 


ELECTROMAGNETIC    INDUCTION.  337 

the  fixed  medium  and  field.  This  would  be  expected  from  the 
fact  that  the  motional  intensity  can  always  be  calculated  from  the 
motion  of  the  tubes  of  induction  with  reference  to  the  conductor, 
and  the  fact  that  the  forcive  upon  the  conductor  is  in  both  cases 
equal  and  opposite  to  the  forcive  upon  the  magnets  or  circuits 
producing  the  field,  so  that  the  work  done  by  or  against  the 
electromagnetic  forces  during  a  given  relative  motion  of  the  -con- 
ductor and  the  magnets  or  circuits  is  in  both  cases  the  same. 
These  intensities  and  e.m.f.s  are  not  intrinsic,  no  energy  being 
transformed  from  mechanical  to  electrical,  or  vice  versa,  in  the 
regions  which  are  their  seats,  since  the  conductor  does  not  move 
(appreciably)  against  or  under  the  action  of  any  force.  Through 
their  agency  electrical  energy  is  transferred  to  or  from  the  con- 
ductor at  rest,  the  transformation  taking  place  at  the  circuit  which 
moves  (in  which  there  is  a  motional  e.m.f). 

Also,  when  the  magnetic  flux  through  a  circuit  changes  owing 
to  any  other  cause  than  the  relative  motion  of  the  circuit  and  a 
magnetic  field,  as  when  the  current  in  the  given  circuit  itself  (if 
a  conductor)  or  the  current  in  a  neighboring  circuit  varies,  the 
tubes  of  magnetic  induction  must  be  conceived  to  move  inward 
or  outward  across  the  circuit,  since  all  tubes  of  magnetic  induc- 
tion are  continuous  or  closed,  and  since  each  tube  has  the  same 
strength  throughout  its  length.  Hence  the  change  of  the  mag- 
netic field  in  this  manner  would  be  expected  to  develop  intensi- 
ties and  e.m.f.s  similar  to  those  developed  by  the  relative  mo- 
tion of  a  conductor  and  magnets  or  circuits  carrying  steady 
currents ;  and  this  expectation  is  fully  confirmed  by  experi- 
ment. These  e.m.f.s  are  not  intrinsic,  energy  being  merely 
transferred,  not  transformed,  through  their  agency,  and  all 
the  (electrical)  energy  so  transferred  coming  originally  (by  trans- 
formation) from  the  intrinsic  e.m.f.s  in  the  circuit,  or  one  of  the 
circuits. 

The  intensities  and  e.m.f.s  considered  in  this  section  are  called 
induced  electric  intensities  and  e.m.f.s,  although,  as  stated  in  §4, 
the  same  term  is  very  generally  applied  to  the  motional  intensity 


338          ELEMENTS    OF   ELECTROMAGNETIC   THEORY. 

and  e.m.f.  also.     When  the  intensity  is  not  intrinsic,  E  should 
be  substituted  for  e  in  (5)  and  (50). 

Satisfactory  direct  experiments  upon  the  induced  intensity  in 
insulators  have  not  yet  been  made ;  but  the  very  important  con- 
sequences of  assuming  that  the  results  established  by  experiment 
for  conductors  apply  to  insulators  as  well  are  justified  by  their 
agreement  with  experiment.  See  Chapter  XVI. 

Lenz's  Law.  The  general  form  of  Lens's  law,  which  is  only  a 
particular  case  of  the  law  of  the  conservation  of  energy,  is  as 
follows  :  Whenever  an  e.m.f.  is  induced  in  any  body,  either  a  con- 
ductor or  a  dielectric,  by  a  variation  of  the  magnetic  field  or  by 
relative  motion  between  the  body  and  magnets  or  circuits  tra- 
versed by  electric  currents,  the  e.m.f.  has  such  a  direction  that 
in  the  resultant  magnetic  field  the  variation  of  the  field,  or  the 
motion,  which  produced  the  e.m.f.  is  opposed. 

Thus  when  a  wire  is  moved  in  a  magnetic  field,  the  field  is 
strengthened  on  the  side  toward  which  the  wire  is  moving  and 
weakened  on  the  other  side,  since  the  force  on  the  wire  is  from 
the  stronger  to  the  weaker  part  of  the  resultant  field  (§  3,  XII.). 
This  gives  the  direction  of  the  motional  e.m.f.  at  once ;  for  it 
gives  the  direction  around  the  wire  of  the  lines  of  intensity  it 
produces,  and  this  direction  bears  a  definite  relation  (§  i,  XIII.) 
to  the  direction  of  the  current  developed  by  the  motional  e.m.f., 
which  is  the  direction  of  the  e.m.f.  itself. 

6.  The  Second  Law  of  Circuitation.  Integral  Form.  Consider 
any  closed  curve,  or  circuit,  in  a  conductor  or  dielectric  traversed 
by  a  magnetic  field.  Let  the  positive  direction  around  the  cir- 
cuit be  chosen  arbitrarily,  and  the  positive  direction  through  the 
circuit  according  to  the  convention  of  §  2,  XII.,  the  first  direc- 
tion being  related  to  the  second  as  the  direction  of  rotation  to 
the  direction  of  translation  of  a  right-handed  screw.  Then,  if  the 
circuit  moves  relatively  to  the  magnetic  field  in  any  manner,  or  if 
the  field  varies  in  any  manner,  or  if  both  changes  occur  together, 
fdLM(B  sin  0)(u  sin  /3)  taken  in  the  positive  direction  once  around 


ELECTROMAGNETIC   INDUCTION.  339 

the  circuit  evidently  denotes  the  rate  at  which  the  magnetic  flux 
in  the  positive  direction  through  the  circuit  is  diminishing.  If  we 
assume  that  electromotive  forces  are  induced  in  insulators  accord- 
ing to  precisely  the  same  laws  as  in  conductors,  it  follows  that 
when  the  magnetic  flux  through  any  circuit  changes  owing  to 
any  causes  an  e.m.f.  is  induced  around  the  circuit  equal  to  the 
rate  at  which  the  magnetic  flux  through  the  circuit  is  decreasing. 
That  is,  if  4>  denotes  the  magnetic  flux  through  the  circuit  (posi- 
tive when  in  the  positive  direction)  and  M*  the  e.m.f.  around  the 
circuit  (positive  when  in  the  positive  direction),  at  the  time  / 

(8) 


which  is  the  second  law  of  circuitation  (in  its  integral  form).     The 
e.m.f.  may  be  wholly,  or  only  partially,  or  not  at  all,  intrinsic. 

As  an  immediate  deduction  from  (8)  it  follows  that  the  e.m.f. 
induced  in  a  coil  of  wire  through  which  the  coil  flux  changes  at 
the  rate  dNjdt  is 

^  =  _  dN\dt  (So) 

The  relative  directions  of  induced  electromotive  force  and 
change  of  magnetic  flux,  as  given  by  (8)  and  (So),  can  also  be 
obtained  immediately  from  Lenz's  law. 

E.M.F.  of  Self  Induction.  Thus  when  the  current  7  in  an  iso- 
lated coil,  of  inductance  L,  increases  at  the  rate  dlfdt,  and  the 
coil  flux  therefore  at  the  rate  dNjdt=d(Lr)jdt,  an  e.m.f. 
—  d(LI}dt  is  induced  in  the  coil.  Thus  the  change  of  the  cur- 
rent or  of  the  magnetic  flux  is  opposed  by  the  induced  e.m.f. 

E.M.F.  of  Mutual  Induction.  Also,  if  the  current  /x  in  one  (i) 
of  two  coils,  with  mutual  inductance  M,  increases  at  the  rate 
dfjdt,  and  therefore  the  coil  flux  -W12  =  MI^  through  the  other 
coil  (2)  due  to  the  first  coil  at  the  rate  dNujdt  =  d(MI^jdt,  an 
e.m.f.  —  d(MI^)jdt  is  induced,  owing  to  this  change  of  flux,  in 
coil  2  (in  addition  to  the  e.m.f.  —  d(L2I^fdt\  At  the  same  time 
there  is  induced  in  coil  I  the  e.m.f.  —  \_d(LJ^ldt  +  d(MI2)/dt] . 


340  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

7.  Differential  Form  of  the  Second  Law  of  Circulation  for  Media 
at  Eest.  The  Curl  of  a  Vector.  Curl  E.  Consider  an  infinites- 
imal plane  circuit  of  area  dS  in  a  changing  magnetic  field  in  a 
medium  at  rest.  Let  B  denote  the  induction  at  dS  and  dB  its 
increase  (in  magnitude  and  direction)  in  the  time  dt.  The  line 
integral  of  induced  electric  intensity,  £,  around  the  edge  of  dSy 
viz.,  \  E  cos  6  dL,  is  evidently  a  maximum  when  dS  is  perpen- 
dicular to  dB,  in  which  case 

f£  cos  0  dL  =  -  d<$>jdt  =  -  dBjdt-dS 

Hence  the  maximum  e.m.f.  around  the  edge  of  dS  per  unit  area  is 
f£  cos  0  dLjdS  =  -  dBjdt 

Now  the  maximum  line  integral  of  a  vector  per  unit  area 
around  the  edge  of  an  infinitesimal  circuit  at  a  point  is  called  the 
curl  of  the  vector  at  the  point.  Hence  the  above  equation  may 

be  written  ,   ^  J-DIJ*  <  \ 

curl  E  =  —  dB  jdt  (9) 

Thus  curl  E  is  a  vector  with  the  magnitude  and  direction  of 
—  dB  jdt.  The  three  components  of  curl  E  along  the  rect- 
angular coordinate  axes  of  X,  Y,  and  Z  are 


(10) 


Cartesian  expressions  for  these  components  will  be  developed 
in  §4,  XVI. 

E  in  the  above  equations  denotes  the  (non  -intrinsic)  induced 
electric  intensity,  and  does  not  include  the  motional  intensity  or 
any  other  intrinsic  intensity,  or  any  intensity  connected  with 
electric  charges,  true  or  fictitious,  if  present.  Since,  however, 
the  line  integral  of  the  latter  intensity  around  any  closed  circuit 
is  zero,  its  curl  is  zero,  and  (9)  will  remain  true  if  E  is  taken  to 
denote  the  vector  sum  of  the  induced  intensity  and  the  intensity 
(whose  curl  is  zero)  due  to  the  presence  of  charges,  true  or 


ELECTROMAGNETIC    INDUCTION. 


341 


fictitious.     The  second  law  of  circuitation  will  be  generalised  in 
Chapter  XV. 

8.  The  Absolute  Determination  of  a  Kesistance  by  Lorenz's 
Method.  A  circular  coil  C,  Fig.  1 10,  mounted  with  the  planes 
of  its  turns  in  the  magnetic  meridian,  is  traversed  by  a  constant 
current  /,  the  resistance  R  to  be  determined  being  included  in 
the  circuit.  With  its  axis  coincident  with  that  of  the  coil,  a  cir- 
cular metallic  disc  D  rotates  with  a  constant  angular  velocity, 


Fig.  110. 

making  /  revolutions  per  second.  Wires  bearing  upon  the  edge 
and  axle  of  the  disc  connect  it  through  a  galvanometer  G  with 
the  terminals  of  the  resistance  R. 

Let  M  denote  the  magnetic  flux  across  the  disc  between  its 
edge  and  the  edge  of  its  axle  per  unit  current  in  the  coil.  Then 
the  magnetic  flux  cut  across  by  every  radius  of  the  disc,  between 
axle  and  circumference,  in  one  complete  revolution  is  MI  (pro- 
vided there  is  no  current  in  the  disc).  Hence  an  e.m.f.  is  devel- 
oped along  every  radius  of  the  disc  equal,  from  axle  to  circum- 
ference or  from  circumference  to  axle,  to  Mpl.  The  connections 
from  the  disc  to  the  terminals  of  R,  for  a  given  direction  of  rota- 
tion of  the  disc,  are  so  arranged  that  the  induced  e.m.f.  Mpl  and 
the  e.m.f.  RT  are  in  opposition  through  the  galvanometer ;  then 
the  resistance  or  the  speed  of  the  disc  is  adjusted  until  the  gal- 
vanometer shows  no  deflection  when  the  galvanometer  circuit  is 
either  open  or  closed.  Then 

MpI=RI 


342          ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

and  R  =  Mp  (n) 

/  can  be  measured  directly,  and  Mean  be  calculated  from  the 
linear  dimensions  and  number  of  turns  of  the  coil.  Hence  R  can 
be  determined  in  absolute  measure. 

If  C  is  a  long  solenoid,  and  if  D  is  placed  well  within  C, 

M  •=  tip  X  Trr2  —  »//.  x  7rr/2  =  H/JLTT^  —  r/2) 


where  ^  denotes  the  number  of  turns  of  wire  per  cm.  upon  the 
solenoid,  and  r  and  rf  denote  the  radii  of  the  disc  and  its  axle. 

The  method  serves  also  for  the  comparison  of  low  resistances, 
in  which  case  J/need  not  be  known. 

9,  Induction  Discharge  Through  a  Circuit.  When  the  coil  flux 
through  a  coil  of  resistance  R  changes  from  one  value  N^  to 
another  value  N2  as  the  time  changes  from  tl  to  t$  an  electric 
charge 


due  to  electromagnetic  induction,  circulates  in  the  positive  direc- 
tion around  the  circuit.  Here  /  denotes  that  part  of  the  current, 
and  M/*  that  part  of  the  electromotive  force,  in  the  circuit  due  to 
the  change  of  flux. 

Thus  q  depends  wholly  upon  the  resistance  and  the  total 
change  in  the  flux,  and  not  at  all  upon  the  time  or  the  way  in 
which  this  change  takes  place. 

10.  The  Electrokinetic  Energy  of  the  Field  of  an  Isolated 
Circuit.  The  magnetic  energy  residing  in  the  field  of  a  single 
coil  or  circuit  whose  coil  flux  is  N  when  its  current  is  7  is 


w=  \IN=  \LP  =  \N*IL  (13) 

provided  that  L  —  N\I=  constant,  that  is,  provided  ft  is  con- 
stant. 

For,  if  no  energy  is  dissipated,  the  energy  of  the  field  is  equal 
to  the  work  done  against  the  counter  e.m.f.  —  dNjdt  in  increas- 


ELECTROMAGNETIC    INDUCTION.  343 

ing  the  current  from  o  to  /  or  the  coil  flux  from  o  to  N.  If  N 
and  /  denote  also  the  instantaneous  values  of  the  coil  flux  and 
current,  the  work  done  against  the  e.m.f.  —  dNjdt,  or  the  energv 
stored  in  the  field,  in  the  time  dt  is 

dW=  IdNjdt  dt  =  IdN  (14) 

Hence 


CidN=  Fudi= 

Jo  Jo 


which  is  identical  with  (13). 

The  same  result  follows  from  §  19,  XL,  which  gives 


etc. 

W  is  called  the  electrokinetic  energy  of  the  field.  A  mechan- 
ical conception  of  this  energy  is  given  in  §  1  1  ,  B. 

11.  A.  Mechanical  Analogues  of  L,  I,  N,  ¥,  dN/dt,  and  W. 
(i)  Let  L,  7,  and  N  '=  LI  denote  the  moment  of  inertia,  angular 
velocity,  and  angular  momentum,  respectively,  of  a  rigid  body  B 
about  a  given  axis.  If  the  angular  velocity  /  is  increased  at 
the  rate  dljdt,  and  the  angular  momentum  at  the  rate  dNjdt 
=  d(LI)ldt,  the  increase  will  be  opposed  by  a  torque  of  inertia 
equal  to  M*  =  —  dNjdt.  To  overcome  this  torque,  that  is  to  in- 
crease the  velocity  or  momentum,  an  equal  and  opposite  torque 
+  dNjdt  must  be  applied.  The  rate  at  which  work  is  done  in 
increasing  the  velocity  or  momentum  is  dWjdt  =  IdNjdt^  and 
the  total  work  done  in  increasing  the  velocity  from  o  to  /,  or  the 
momentum  from  o  to  N  =  LI,  is 


W  =  ffdN  =$LIdI  =  \IN  =  \LP 

which  is  the  kinetic  energy  of  B  when  its  angular  velocity  is  7. 
(2)  Let  Z-,  7,  and  N  =  LI  denote  the  mass,  linear  velocity, 
and  momentum,  respectively,  of  an  incompressible  liquid  flowing 
in  a  closed  pipe  of  constant  cross  -section.  If  the  velocity  7  is 
increased  at  the  rate  dljdt,  and  the  momentum  at  the  rate 


344          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

dNjdt,  the  increase  will  be  opposed  by  a  force  of  inertia  equal 
to  —  dNjdt  =  M*.  To  overcome  this  inertia,  that  is,  to  increase 
the  velocity  or  momentum,  an  equal  and  opposite  force  -f  dNjdt 
must  be  applied.  The  rate  at  which  work  is  done  in  increasing 
the  momentum  is  dWjdt=  IdNjdt,  and  the  total  work  done  in 
increasing  the  momentum  from  o  to  TV  =  LI  is 


which  is  the  kinetic  energy  of  the  liquid  when  its  velocity  is  /. 

Neither  of  these  analogues  is  at  all  complete.  Thus  the 
energy  in  (i)  resides  wholly  in  the  rigid  body  and  the  energy 
in  (2)  resides  wholly  inside  the  pipe,  while  the  energy  of  an  elec- 
tromagnetic field  may  reside  almost  wholly  in  the  dielectric  sur- 
rounding the  conductor. 

B.  Mechanical  Conception  of  the  Magnetic  Field.  Let  //-,  H, 
and  B  =  pH  denote  the  moment  of  inertia  per  unit  volume,  angu- 
lar velocity,  and  angular  momentum  per  unit  volume,  respectively, 
at  a  point  P  in  an  incompressible  medium  in  a  given  type  of  rota- 
tory motion.  If  the  angular  velocity  H  is  increased  at  the  rate 
dHjdt,  and  the  angular  momentum  per  unit  volume  therefore  at 
the  rate  dBjdt  =  pdHjdt  the  increase  will  be  opposed  by  a  torque 
of  inertia  equal,  per  unit  volume  at  P,  to  curl  E  =  —  dBjdt  (E 
being  the  force  per  unit  area  acting  tangential  to  the  surface  of 
the  rotating  element). 

Adhering  to  the  fundamental  conception  of  the  electric  field, 
§§  13-14,  I.,  according  to  which  E  is  a  kind  of  shearing  stress 
between  positive  and  negative  aether  cells,  we  shall  therefore 
assume  that  at  any  point  of  a  magnetic  field  in  free  aether  the 
positive  and  negative  cells  are  rotating  with  equal  angular  speeds 
in  opposite  directions  (unlike  cells  being  in  contact  or  geared  to- 
gether) the  axes  of  rotation  being  parallel  to  the  direction  of  the 
intensity  at  the  point,  and  the  direction  of  rotation  of  the  positive 
cells  being  related  to  the  direction  of  the  intensity  as  the  rotation 
to  the  translation  of  a  right-handed  screw.  We  shall  assume 


ELECTROMAGNETIC    INDUCTION.  345 

the  magnetic  intensity  H  to  be  the  angular  velocity  of  the  posi- 
tive cells,  equal  and  opposite  to  the  angular  velocity  of  the 
negative  cells.  We  shall  further  assume  p  to  be  the  sum  per 
unit  volume  of  the  moments  of  inertia  of  the  cells.  Then  \B 
=  \^H  will  be  the  sum  of  the  angular  momenta  of  the  positive 
cells  per  unit  volume,  and  —  \B  —  —  \pH  will  be  the  sum  of 
the  angular  momenta  of  the  negative  cells  per  unit  volume.  The 
total  kinetic  energy  per  unit  volume  will  be  J(J//J7  x  H)  -f 
J(— J/A//X  —H}=lfiH2.  The  centrifugal  force  arising  from 
the  rotation  of  the  cells  will  account  for  the  pressure  normal  to 
the  lines  of  magnetic  intensity,  and  therefore  for  the  tension  along 
the  lines  of  intensity,  and  these  tensions  and  pressures  will  account 
for  the  mechanical  forces  upon  magnets,  for  Ampere's  law,  etc. 
For  a  fuller  account  of  this  conception  and  its  application  to  nu- 
merous electrical  phenomena  see  Lodge's  Modern  Views  of  Elec- 
tricity, and  an  article  by  W.  S.  Franklin,  Physical  Review,  Vol.  4. 

12.  Electrokinetic  Energy  Density  in  a  Medium  With  Constant 
Inductivity  (ft).     Fig.  1 1 1  represents  a  plane  section  through  a 


coil  RR  with  n  turns  (n  =  3  in  the  figure),  an  equipotential  sur- 
face S,  and  a  tube  of  induction  A  BCD  A  threading  the  circuit  RR 
and  cutting  out  from  the  equipotential  an  element  of  area  dS. 


346          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

If  /  denotes  the  current  through  the  coil,  the  m.m.f.  around 
the  tube  ABCDA  is 


The  magnetic  flux  across  the  equipotential  5  is 

Njn  =$BdS 
Hence  the  energy  of  the  magnetic  field  is 

W  =  \IN=  \nIN\n  =  \§HdL$BdS 

\BHdr 


the  integration  being  extended  throughout  the  magnetic  field. 
The  energy  per  unit  volume  is  therefore 


T=  dWjdr  =  \BH  =  |/z7/2  =  *&/p  (16) 

as  otherwise  proved  in  §  18,  XI. 

For  the  work  done  during  the  process  of  magnetisation  when 
/i  is  a  function  of  //"see  §  18,  XL,  or  §  29. 

13,  The  Electrokinetic  Energy  of  the  Field  of  Two  or  More 
Circuits  in  a  Medium  of  Constant  Inductivity  (JJL).  Consider  first 
two  circuits,  I  and  2,  Fig.  1  12,  with  nv  and  n2  turns  and  currents 
7X  and  72,  respectively.  Let  Lv  L2,  M2V  and  M12  denote  the  self 
inductances  and  coefficients  of  induction  (§  3,  XIII.)  of  the  two 
circuits.  Let  Hv  denote  the  magnetic  intensity  at  any  point  P 
due  to  the  current  7X  alone,  H2  that  due  to  the  current  /2  alone, 
and  H  the  resultant  intensity  due  to  7t  and  72  together.  Then 

H*  =  ff*  +  7/22  +  2H1H2  cos  012 

if  B12  denotes  the  angle  between  7^  and  772.     The  total  energy 
in  the  magnetic  field  is,  by  §  12, 


cos 
the  integrations  extending  throughout  the  magnetic  field. 


ELECTROMAGNETIC    INDUCTION.  347 

The  first  and  second  integrals  are  equal,  respectively,  to 
iVi2  and  iA/22>  by  §§  10  and  12. 

To  evaluate  the  third  integral,  consider  a  magnetic  equipoten- 
tial  surface  5  for  the  field  of  7X  alone  drawn  through  P,  and  T,  a 
tube  of  induction  of  this  field  enclosing  P  and  cutting  out  from  vS 


Fig.  112. 


an  element  of  area  dS.  With  N,  the  normal  to  dS,  H2  and  B^ 
make  the  angle  012.  The  integral  §  pH^H^  cos  012  dr  may  evi- 
dently be  written 

cos  0    dSdL  =HdLH  cos  0   dS 


in  which  the  first  integration  extends  around  the  tube  T  and  the 
second  over  the  surface  5.     Now,  by  (22),  XII., 


and,  by  the  definitions  of  magnetic  flux  and  coil  flux, 


Hence       ff^Lpff,  cos 

and  W=  fenHV-r  =  ±LJ?  +  \LJ}  +  MJ&  (17) 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

If,  in  evaluating  the  third  integral,  we  interchange  the  roles  of 
circuits  I  and  2,  we  obtain 


(if) 
Since  the  two  expressions  for  ^Fare  equal, 

Ma-Ma  (18) 

as  stated  in  §  3. 

The  term  M^I^I^  =  M^J^I^  is  called  the  mutual  energy  of  the 
two  coils  or  fields.  In  the  same  way,  in  general,  the  expression 
NyJi  denotes  the  mutual  energy  of  the  fields  I  and  2. 

In  exactly  the  same  manner  it  may  be  shown  that  if  we  have 
any  number  of  circuits  1  ,  2,  •  •  •  ,  n  with  currents  Iv  /2,  •  •  •  ,  /n, 
inductances  Lv  L2,  •  •  •  ,  Zn,  and  coefficients  of  mutual  induction 
M12,  Miy  -  •  •  ,  M2y  Mw  -  •  •  ,  etc.,  then  the  electrokinetic  energy  is 

W=\Lj;+\L£  +  ...  +  \LJt 

+  j/12//2  +  j/13//3  +  .  .  .  +  J4/2/3  +  •  •  •     (19) 

14.  The  Coefficient  of  Self  Induction  of  a  Coil  is  Proportional 
to  the  Square  of  its  Number  of  Turns,  the  dimensions  of  the  coil 
being  kept  constant.  For  if  <£  denotes  the  average  flux  through 

a  single  turn  of  the  coil, 

3>  =  knl 

where  k  is  a  constant.     Hence 

=  kn2  (20) 


15.  The  Coefficient  of  Mutual  Induction  of  Two  Coils  is  Pro- 
portional to  the  Product  of  the  Numbers  of  Turns  in  the  Two  Coils, 
the  dimensions  and  positions  of  the  coils  remaining  constant.  For 
if  <I>12  denotes  the  average  flux  through  a  single  turn  of  coil  2 
due  to  a  current  /L  in  coil  I, 

3>12  =  kn^ 
where  k  is  a  constant.      Hence 


ELECTROMAGNETIC    INDUCTION.  349 

16.  The  Coefficient  of  Self  Induction  of  a  Coil  is  Proportional  to 
its  Linear  Dimensions,  the  shape  of  the  coil  and  its  number  of 
turns  remaining  constant.  To  prove  this,  consider  a  coil  A,  and 
another  coil  A'  with  linear  dimensions  k  times  as  great.  If  P 
and  P'  are  corresponding  points,  5  and  S'  corresponding  equi- 
potential  surfaces,  dS  and  dS'  corresponding  elements  of  area, 
B  and  Bf  corresponding  inductions,  <I>  and  <£>'  corresponding 
average  fluxes  through  a  single  turn  for  the  same  current  7,  and 
L  and  L'  the  corresponding  inductances  for  the  two  coils,  then 
B'=ijkBt  by  §  n,  XII.,  and  dSf  =  PdS.  Hence 


and  L'  =  n<$>'  !I=kn<$>lI=kL  (22) 

which  was  to  be  proved. 

In  like  manner  it  may  be  shown  that  the  coefficient  of  mutual 
induction  of  two  coils  is  proportional  to  the  linear  dimensions, 
the  relative  dimensions  and  distances  retaining  the  same  ratios. 

17.  Energy  Definitions  of  Self  and  Mutual  Inductance.  From 
(13),  §  10,  the  coefficient  of  self  induction  of  a  coil  or  other  con- 
ductor may  be  defined  as  the  ratio  of  twice  the  energy  of  its 
magnetic  field  to  the  square  of  its  current.  Thus 

L  =  2  IV/I2  =JV//VT//2  (23) 

This  definition  is  not  identical  with  that  of  §  2  unless  JJL  is  con- 
stant. It  is  often  more  convenient  than  the  previous  definition 
in  getting  the  inductance  of  non-linear  circuits  or  conductors. 

In  like  manner  the  coefficient  of  mutual  induction  of  two  cir- 
cuits or  conductors  may  be  defined  as  the  ratio  of  their  mutual 
magnetic  energy  to  the  product  of  their  currents,  by  §  13.  Thus 

Ma  ~(WU  =  MM  I  /A  =  (fpffft  cos  VT)///,    (24) 

From  the  definitions  just  given  the  relations  proved  in  §§  14- 
16  on  the  basis  of  the  earlier  definitions  may  be  readily  estab- 


350          ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

lished.  Thus  if  the  number  of  turns  of  a  coil  is  altered  in  any 
ratio  n,  the  dimensions  and  the  current  remaining  unaltered,  the 
magnetic  intensity  H  in  every  element  of  volume  dr  will  be 
altered  in  the  same  ratio.  Hence  L  =^^H2drjP  will  be  altered 
in  the  square  of  the  same  ratio.  The  other  relations  may  be 
established  in  like  manner. 

18.  The  Inductance  of  Any  Number  of  Coils  Connected  in  Series. 

If  all  the  coils  of  §  1 3  are  connected  in  series,  fl  =  72  =•••  =  /. 
Hence,  if  L  denotes  the  inductance  of  the  system  in  series, 

... 

If  the  individual  coils  are  so  constructed  or  so  far  apart  that 
all  the  mutual  inductances  vanish, 

L  =  L,  +  L2+...  +  Ln  (26) 

The  principle  of  (26)  is  commonly  applied  in  the  construction 
of  standards  of  inductance  variable  by  fixed  amounts,  and  that 
of  (25)  in  the  construction  of  continuously  adjustable  standards 
of  inductance.  For  this  purpose  two  circular  coils  with  induc- 
tances L}  and  L2  are  mounted  (i)  with  their  axes  coincident  and 
the  distance  between  their  centers  adjustable,  or  (2)  with  their 
centers  coincident  and  the  angle  between  the  planes  of  their  turns 
adjustable.  Thus  the  mutual  inductance  M  is  adjustable,  and 
therefore  the  resultant  self  inductance, 

L  =  L,  +  L,  +  2M  (27) 

M  is  positive  or  negative  according  as  the  magnetic  flux  due 
to  one  coil  threads  the  other  in  the  positive  or  negative  direction, 
the  direction  of  the  current  around  each  circuit  being  chosen  as 
the  positive  direction  around  the  circuit. 

It  will  be  shown  in  §  44,  that  a  condenser  of  capacity  5  con- 
nected in  series  with  a  coil  of  inductance  L  produces  the  same 
effect  in  the  case  of  harmonic  alternating  currents,  as  an  induc- 
tance L  —  i  jSp2,  where  /  =  2ir  x  the  frequency  of  the  current. 


ELECTROMAGNETIC    INDUCTION.  351 

19.  1^2  —  M2  is  not  Less  than  Zero,     Let  LL  and  Z2  denote 
the  self  inductances  of  two  coils,  and  M  their  mutual  inductance. 
Then 

L^-M^o  (28) 

For  the  electrokinetic  energy  of  the  field,  given  by  (17),  is  a 
signless  or  positive  quantity,  whatever  the  values  of  the  currents  ; 
.and  (28)  is  the  condition  that  this  expression  may  never  be  less 
than  zero,  whatever  the  values  of  the  currents. 

The  proposition  may  also  be  demonstrated  as  follows.  In  the 
nomenclature  of  §  15, 

A  =  *i*,/4  A  =  «,*,/'„  and  M=  «,*„//,  =  *,*„//; 
Hence 

AA  -  M*  =  «,«,///,  .(*,*,  -  4>21*I2) 

=  «,«,*,*,///,  -(I  -  *„/*,.  *„/*,) 

which  is  greater  than  zero,  or  equal   to  zero,  since  each  of  the 
last  fractions  is  less  than  unity,  or  unity. 

The  sign  of  equality  holds,  or  ZXZ2  =  M. 2,  only  when  all  the 
magnetic  flux  threads  every  turn  of  both  coils  (4>12  =  <I>21  ==  <X>X 
=  <I>2),  as  when  the  coils  are  toroidal  and  uniformly  wound  on 
the  same  core. 

20,  The  Inductance  of  a  Uniformly  Wound  Solenoid  and  Its 
Electrokinetic  Energy.    The  flux  through  each  turn  of  the  infinite 
solenoid  of  §  19,  XII.,  is  <£  =  pHS  =  pSnf,  and  the  coil  flux 
through  the  nA  turns  in  a  length  A  of  the  solenoid  is  N=  S^nlnA 
=  pSIn2A.    Hence  the  inductance  of  a  length  A  of  the  solenoid  is 

L  =  N/f=fjLSn2A  (29) 

The  same  result  may  be  obtained  otherwise  thus  :  The  mag- 
netic energy  per  unit  volume  within  the  solenoid  is  J/^//"2  =  ^/-i/z2/2, 
and  there  is  no  energy  outside  the  coil.  The  volume  of  a  length 
A  of  the  solenoid  is  SA,  and  the  energy  contained  in  this  vol- 
ume is 


352          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

W=  lfiH2SA  =  ifin2f2SA  (30) 

Hence  Z  =  2  ^//2  =  fiSn^A  (3  1  ) 

Since  the  magnetic  field  is  confined  wholly  to  the  region 
within  the  coil,  the  inductivity  of  the  surrounding  medium  does 
not  affect  the  inductance. 

If  the  length,  A,  of  a  finite  solenoid,  §  20,  XIL,  is  great  in 
comparison  with  its  cross-section  S,  the  intensity  within  the 
solenoid  is  sensibly  uniform  and  equal  to  H  =  nl  except  near  the 
ends,  where  it  is  weaker,  and  the  external  field,  except  near  the 
ends,  is  very  weak.  Hence  the  inductance  of  the  solenoid  is 
approximately  that  given  by  (31),  the  approximation  becoming 
more  exact  as  the  length  of  the  solenoid  increases,  since  the 
internal  energy  increases  almost  proportionally  to  the  length 
(slightly  faster)  and  the  external  energy  increases  but  slightly 
with  increase  of  length  (see  §  8,  VI.).  The  inductivity  of  the 
external  medium  affects  the  inductance  only  slightly,  inappreci- 
ably when  the  solenoid  is  very  long. 

The  permeance  of  the  solenoid  is  sensibly 

P=  <£/0  =  nHSjHA  =  fji  S/A  (32) 

21.  The  Electrokinetic  Energy  Contained  in  an  Isolated  Circular 
Cylindrical  Conductor.  (1)  Solid  Cylinder.  The  energy  con- 
tained in  an  elementary  cylindrical  shell  of  length  A,  radius  r, 
and  thickness  dr  is 

dW 


Within  the  conductor  at  a  distance  r  from  its  axis 


if  R  denotes  the  radius  of  the  wire  and  /  the  current.      Hence 

W=    fdW  =  nf2A/47rR4  f  i*dr  =  pI2A/i67i         (33) 


Thus  the  energy  within  the  wire,  for  a  given  current,  is  inde- 
pendent of  the  radius  of  the  wire  and  is  proportional  to  its  indue- 


ELECTROMAGNETIC    INDUCTION.  353 

tivity.     The  external  energy  is  less  the  greater  the  radius  of  the 


wire. 


?  -  Rff  f  *V  - 
Jn, 


(2)  Hollow  Cylinder.  Let  the  inner  and  outer  radii  be  denoted 
by  ^  and  R2,  respectively.  Then,  within  the  shell,  at  a  point 
distant  r  from  the  axis 


Hence      W  = 


(34) 

In  this  case  the  electrokinetic  energy  depends  on  the  ratio  of 
R2  to  Rv  being  greater  the  greater  this  ratio,  and  approaching 
zero  as  the  ratio  approaches  unity.  The  correctness  of  the  last 
statement  is  easier  to  see  from  the  following  considerations  than 
from  (34).  For  a  given  value  of  the  current  and  R2,  the  external 
field  is  wholly  independent  of  the  magnitude  of  Rr  The  internal 
intensity  steadily  decreases  from  the  outer  to  the  inner  surface, 
being  equal  to  the  external  intensity  at  the  outer  surface  and  to 
zero  at  the  inner  surface.  The  volume  of  the  shell  approaches 
zero  as  R2  —  Rl  approaches  zero.  Hence  the  internal  energy, 
which  is  equal  to  the  volume  of  the  shell  times  the  'average 
energy  density,  approaches  zero  as  R2jRl  approaches  unity  or  as 
R2  —  Rl  approaches  zero,  while  Rl  or  R2  remains  constant. 

22.  The  Electrokinetic  Energy  and  Inductance  of  a  Cable  con- 
sisting of  a  circular  cylindrical  core  of  inductivity^  /JL  and  radius 
R  and  a  coaxial  circular  cylindrical  shell  of  the  same  inductivity 
and  internal  and  external  radii  R^_  and  R2,  the  inductivity  of  the 
intervening  dielectric  being  /n. 

The  magnetic  energy  contained  in  a  length  A  of  the  core  (sup- 
posed solid)  is 


354          ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

The  magnetic  energy  in  a  length  A  of  the  dielectric  is 
(b)  =^'H2A27rrdr  =  fjLff2A/4ir  •  log  RJ  R 


The  intensity  within  the  shell  at  a  point  distant  r  from  the 
axis  is 


_  R*ft  (3  5) 

Hence  the  magnetic  energy  within  the  shell  is 
(c)  =  p 


The  total  magnetic  energy  in  the  length  A  of  the  cable  is 
•     W=(a)  +  (6)  +  (f)          :•-.  .;n.    ..'.v 
The  inductance  of  a  length  A  of  the  system  is  therefore 
L  =  2  WjP  =  fjiA/STT  +  /AM/27T  •  log  ^/tf  +  M/STT  -  [(R* 
-  3^22)/(^2  -  ^2)  +  4^24/(^22  -  W  •  log  *,/*J 

If  the  outer  shell  is  extremely  thin,  the  third  term  becomes 
negligible,  and  we  have,  very  approximately, 

L  =  pA/Sir  +  p'A/2ir  •  log  RJR  (37) 

If  the  core  of  the  cable  is  a  very  thin  hollow  cylinder,  instead  of 
a  solid  cylinder,  the  first  term  also  vanishes  approximately,  and 

L  =  p'A/2ir  •  log  RJR  (38) 

(38)  is  rigorously  true  when  the  conducting  shells  are  infinitely 
thin,  or  when  the  conductors  are  perfect  conductors,  both  of 
course  ideal  cases. 

23.  The  Magnetic  Energy,  Inductance,  etc.,  of  a  Rectangular 
Toroid  (Fig.  103,  §  22,  XII.).  Let  the  uniform  thickness  of  the 
toroid  parallel  to  the  axis  of  revolution  be  denoted  by  b,  and  the 
inner  and  outer  radii  by  Rl  and  Rv  and  let  the  whole  number  of 
turns  in  the  coil  be  denoted  by  n. 

Then  the  m.m.f.  along  a  closed  line  of  intensity  is 

'  fi=  nl 


ELECTROMAGNETIC    INDUCTION.  355 

The  intensity  at  a  distance  x  from  the  axis,  when  x  is  greater 
than  R  and  less  than  R^  is 


(39) 
and  the  induction  is 

B  =    JLff  = 


The   magnetic   flux  across  an  elementary  strip  of  length  b 
parallel  to  the  axis,  and  breadth  dx  perpendicular  to  the  axis  is 

Bbdx  =  fjLn&f/27r  •  dxjx 


if  the  strip  is  distant  x  from  the  axis.     The  total  flux  through 
a  single  turn  of  the  coil  is  thus 


r  •  log  RJRl       .    (40) 
and  the  coil  flux  is 

-log  R2/Rl 


whence 

L  =  N/f  =  f*«2£/2ir-log  R2fRl  (41) 

The  permeance  and  reluctance  of  the  magnetic  field  are  given 
by  the  equation 


p  =  i  IR  =  <D/n  =  ^/27r  •  log  RJRI  (42) 

The  electrokinetic  energy  within  the  tore  is 

W=  \LP  =  }IME>  =  ^W2/27r  •  log  R2jRl          (43) 


24.  The  Inductance  and  Electrokinetic  Energy  of  a  System  Con- 
sisting of  Two  Parallel  Circular  Cylindrical  Wires  traversed  by 
the  same  current  in  opposite  directions. 

Let  the  distance  d  between  the  axes  of  the  wires  be  great  in 
comparison  with  R,  the  common  radius  of  the  wires.  Then  the 
energy  contained  in  a  length  A  of  each  wire  is  very  nearly  the 
same  as  if  the  wires  were  infinitely  remote  from  one  another, 
viz., 


356          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

The  energy  within  a  length  A  of  the  dielectric  is  very  ap- 
proximately equal  to  ^<£/,  where  <l>  is  the  magnetic  flux  across 
the  rectangular  area  A(d  —  2R)  connecting  a  length  A  of  the 
wires. 

The  magnetic  induction  B  at  every  point  of  the  area  A  x  (d  — 
2R)  is  normal  thereto.  Let  x  denote  the  distance  of  a  point  in 
this  plane  from  one  of  the  wires,  and  /*'  the  inductivity  of  the 
dielectric  ;  then  the  induction  at  the  point  is 

B  =  /*'//27rr  -f  A<///27r(y  —  x) 
Hence  the  magnetic  flux  through  the  area  is 


<f>  =  B-  Adx  =  n'AIJTr  .  log  [(d  -  R)/R] 

JR 

The  energy  contained  in  the  tubes  crossing  the  area  A(d  — 
2R)  ls  (b]  =  !<!>/=  fji'AS2/27r-  log  \_(d-  R)/R] 

The  total  energy  in  a  length  A  of  the  system  is 
W=  2(0)  4-  (b}  =  M/2/87r  +  v'AS2/27r  •  log  \_(d  -  R)/R]   (44) 
and  the  inductance  of  a  length  A  is 

Z  =  M/4"-  +  pfAI-ir  •  log  [(d-  R)jK\  (45) 

25.  Two  Standard  Mutual  Inductances,  (a)  If  a  coil  (2)  of  n' 
turns  is  wound  around  the  endless  solenoid  (i)  of  §  20,  the  coil 
flux  through  this  outer  coil  (2)  will  be 


and  the  coefficient  of  mutual  induction  of  the  two  coils  will  be 

-WB-A'U//1-A*S»»/  (46) 

If  the  solenoid  has  the  length  2Z  and  a  circular  cross-section 
of  radius  R,  small  in  comparison  with  2Z,  the  field  at  all  points 
of  the  central  portion  is  very  nearly  uniform  and  equal  to  the 
intensity  at  the  center  of  the  axis,  viz., 


ELECTROMAGNETIC    INDUCTION  357 

Hence  if  a  coil  of  nr  turns  is  wound  about  the  central  portion 
of  this  solenoid, 


and  the  coefficient  of  mutual  induction  is 

(47) 


which,  when  L  is  large  in  comparison  with  R,  becomes  practi- 
cally identical  with  (46),  5  being  put  equal  to  irR2. 

(&)  In  the  same  way,  if  a  coil  (2)  of  n'  turns  is  wound  around 
the  closed  coil  (i)  of  §  23,  the  coil  flux  through  this  outer  coil 

be 


N'n'  =  N12  =  pnn'IJ}\2Tr  •  log  RJRl 
and  the  coefficient  of  mutual  induction  of  the  two  coils  will  be 

/2ir-log  R2/Rl 


26.  The  Work  Done  in  Increasing  the  Coil  Flux  Through  a  Coil 
with  a  Constant  Current.  Let  the  constant  current  be  denoted 
by  /.  If  the  coil  flux  is  changing  at  the  rate  dNjdt,  an  e.m.f. 
—  dNjdt  is  developed  tending  to  diminish  the  current,  To  keep 
the  current  constant,  energy  must  be  supplied  (in  addition  to  that 
supplied  when  the  flux  remains  constant)  to  the  circuit  (by  in- 
creasing the  e.m.f.  of  the  battery  or  other  source)  at  the  rate 

dWfdt  =  IdNjdt 

just  sufficient  to  balance  this  induced  e.m.f.  The  work  done  on 
the  circuit,  that  is  on  the  magnetic  field,  while  the  flux  increases 
by  dN  during  the  time  dt  will  be 

dWjdt  dt=dW=  IdNjdt  dt  =  IdN 

Hence  if  the  flux  changes  from  N^  to  Nv  the  magnetic  field 
must  receive  an  increment  of  energy  equal  to 

Wt-W^I(Nt-N^  (47) 

which  is  a  particular  case  of  the  equation  following  (14). 


355         ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

If  N2  —  NI  is  negative,  the  work  W2  —  Wl  is  also  negative,  or 
the  work  is  done  by  the  magnetic  field,  and  its  energy  diminishes 
by  this  amount. 

27.  The  First  Law  of  Circuitation.  The  Magnetomotive  Force 
Around  a  Closed  Path  Linking  with  an  Electric  Current.  Consider 
an  ideal  permanent  flexible  *  magnet  with  concentrated  poles  in 
the  field  of  a  circuit  consisting  of  a  single  turn  of  a  conductor 
traversed  by  a  current  which  is  kept  constant.  Suppose  the 
negative  pole  of  the  magnet  to  remain  fixed  in  position  and  the 
positive  pole  to  be  moved  from  its  initial  position  along  a  closed 
path  linking  once  with  the  circuit  back  to  its  initial  position. 
During  this  process  the  flux  through  the  circuit  changes  by  the 
flux  passing  through  the  body  of  the  magnet  (equal  to  ;;?,  the 
strength  of  its  poles),  and  this  is  the  only  change  in  the  flux 
through  the  circuit  that  occurs.  Since  the  pole  is  in  its  initial 
position,  and  the  current  is  unchanged,  the  energy  of  the  mag- 
netic field  is  unaltered.  Hence  the  work  done  by  the  magnetic 
field  on  the  pole  is  equal  to  the  work  done  upon  the  magnetic 
field  during  the  change  of  flux.  That  is,  if  H  denotes  the  inten- 
sity due  to  the  current  /, 

mfff  cos  6dL  =  /  (N2  -  NJ  =  Im 
Hence 

O  =  fff  cos  OdL  =  /  (48) 

If  there  are  n  turns  to  the  coil,  or  if  the  path  links  n  times  with 

a  coil  of  one  turn, 

H  =  «7 

Thus  the  m.m.f.  once  around  a  closed  path  in  the  positive 
direction  through  a  circuit,  or  in  the  direction  of  the  lines  of 
intensity,  is  equal  to  the  total  current  in  the  positive  direction 
around  the  circuit. 

*Not  essential  but  convenient.  See  Nichols  and  Franklin's  Elements  of  Physics, 
Vol.  II.,  \  124. 


ELECTROMAGNETIC    INDUCTION.  359 

28.  Differential  Form  of  the  First  Law  of  Cireuitation.     Con- 
sider an  infinitesimal  circuit  of  area  dS  in  a  conductor  traversed 
by  a   steady  current  whose   density  at  dS  is  i.     The   current 
across  dS  will  be  a  maximum,  idS,  when  dS  is  turned  with  its 
normal  pointing  in  the  direction  of  i.     In  this  case  the  m.m.f.  in 
the  positive  direction  around  the  circuit  is 

§H  cos  6dL  =  idS 
Hence  the  line  integral,  or  m.m.f,  per  unit  area  is 

curl  77=  i  (49) 

In  the  above  equations  H  denotes  the  magnetic  intensity  of 
the  current,  and  does  not  include  the  field  intensity,  or  intrinsic 
intensity,  connected  with  magnets.  Inasmuch,  however,  as  the 
former  intensity  has  no  curl  (m.m.f.  around  a  closed  path  zero), 
(49)  will  remain  true  if  H  is  taken  to  denote  the  total  intensity 
exclusive  of  the  intrinsic  intensity. 

The  above  relation,  (48)  or  (49),  established  here  for  con- 
duction currents,  is  true  for  the  general  electric  current  (§  7, 
XV.),  and  is  called  \hzfirst  law  of  circulation. 

29.  General  Expression  for  the  Work  Done  in  Magnetisation. 
The  work  per  unit  volume  done  in  changing  the  magnetic  induc- 
tion at  any  point  of  any  medium  from  B^  to  B2  was  shown  in 

§  18,  XL,  to  be 

rs 

HdB  (50) 


dWfdr  =  f * 

*  '2?i 


The  same  result  will  now  be  deduced  by  a  different  process. 

For  convenience  we  shall  make  use  of  a  very  long  solenoid  of 
cross-section  .S  uniformly  wound  with  n  turns  per  cm.,  and  we 
shall  suppose  the  space  within  the  coil  completely  filled  with  a 
homogeneous  isotropic  material. 

The  work  done  in  magnetising  the  core,  in  which  we  shall  first 
suppose  that  no  currents  are  induced  during  the  change  of  mag- 
netisation, is  equal  to  the  work  done  against  the  counter  e.m.f. 


360          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

developed  in  the  coil  by  the  changing  induction.  The  rate  at 
which  work  is  thus  done  upon  a  portion  of  the  core  of  length  L 
and  volume  r  =  SL  is 

dWjdt  =  IdNjdt 

where  N  denotes  the  coil  flux  through  the  portion  of  the  sole- 
noid surrounding  the  volume  considered.  Hence  the  work  done 
in  changing  the  flux  from  N^  to  N2,  or  in  changing  the  induction 
from  B  to  B  is 

Ni 

=T       HdB 


since  nl  '  =  //and  N  =  nLBS.  From  this  equation  (50)  follows 
immediately. 

If  the  core  is  conducting,  currents  (called  eddy  currents  or 
Foucault  currents)  will  be  induced  in  the  core  itself  durine  the 

/  o 

change  of  the  induction,  and  more  work  than  that  given  by  (50) 
will  be  done,  the  extra  energy  going  to  Joulean  heat.  These 
currents  and  the  consequent  dissipation  of  energy  can  be  greatly 
reduced  by  going  through  the  process  slowly,  or  by  dividing  the 
substance  up  into  parts  insulated  from  one  another  in  the  direc- 
tion of  flow  of  the  eddy  currents. 

(50)  gives  the  change  in  magnetic  energy  during  the  change 
of  induction  only  when  there  is  no  dissipation  arising  from  any 
cause. 

(50)  has  been  derived  with  the  aid  of  the  uniform  field  of  a 
very  long  solenoid,  but  since  all  fields  are  uniform  in  their 
infinitesimal  parts,  the  result  is  perfectly  general. 

When  /A  is  independent  of  //or  B,  (50)  reduces  to  (16). 

30.  Electrokinetic  Energy,  Mechanical  Energy,  and  Change  of 
Configuration  of  Circuits.  First  consider  two  circuits,  I  and  2. 
For  the  electrokinetic  energy  we  have 


If,  while  the  currents  are  kept  constant,  the  circuits  so  move 


ELECTROMAGNETIC    INDUCTION.  361 

(or   one  of  the  circuits)  that  M  is  altered   by   an  infinitesimal 
amount  dMt  W  is  altered  by  an  amount 


During  this  motion  the  coil  flux  through  circuit  I  is  increased 
by  the  amount          '..' 

Hence  an  amount  of  work 


is  done  upon  the  magnetic  field. 

In  like  manner,  the  coil  flux  of  circuit  2  is  increased  by  the 
amount 


and  a  second  amount  of  work 


is  done  upon  the  magnetic  field. 

Hence  the  total  work  done  upon  the  magnetic  field  (by  the 
batteries,  in  keeping  the  currents  constant)  is 


while  the  increase  in  its  energy  is  only 


Hence  the  additional  energy  I^I^dM  has  gone  to  increase  the 
mechanical  energy  of  the  system.  If  F  denotes  the  force  acting 
upon  either  circuit  and  tending  to  increase  the  distance  x  between 
the  circuits,  measured  in  its  line  of  action,  the  increase  of  me- 
chanical  energy  is  _  Fdx  = 


the  negative  sign  being  chosen  because  dMis  negative  when  dx 
is  positive.  Hence  the  force  tending  to  increase  the  distance 
between  the  circuits  is 

(51) 


362          ELEMENTS   OF   ELECTROMAGNETIC    THEORY 

If  the  forcive  between  two  circuits  is  a  torque,  instead  of  a 
simple  force,  we  have  for  the  torque  tending  to  increase  the 
angle  6  between  the  planes  of  the  two  circuits, 

T=-IlI2dMld6  (52) 

31.  The  proposition  of  the  last  article  is  a  particular  case  of 
the  following  more  general  theorem  :  If  any  number  of  circuits 
suffer  any  infinitesimal  change  of  configuration,  both  L's  and  M's 
varying  in  the  general  case,  while  the  currents  are  kept  constant, 
the  increase  in  the  electrokinetic  energy,  dWy  is  equal  to  the  in- 
crease in  the  mechanical  energy,  dW  \  while  the  work  done  by  the 
batteries  (over  and  above  that  expended  in  overcoming  resistance 
and  dissipated  in  heat),  dWn  \  is  equal  to 

dW"  =  dW  +  dW  =  2dW=2dW  (53) 

We  proceed  to  establish  this  proposition.  When  the  most 
general  infinitesimal  change  of  configuration  occurs,  the  increase 
in  the  electrokinetic  energy  is,  when  the  currents  are  kept  con- 
stant, 

dw=  I/V 


The  work  done  upon  the  magnetic  field  by  the  batteries  is 
dW"  =  IJNi  +  I2dN2  +  •  •  •  +  IdNn 


Now 


idMln 


Hence 

dW"  =  /.(/ 


I,dL, 


ELECTROMAGNETIC    INDUCTION.  363 


The  difference  between  dW"  =  2dW,  the  energy  supplied  to 
the  magnetic  field,  and  dW,  the  increase  in  its  energy,  must 
equal  the  mechanical  work  dW  done  by  the  field  on  the  circuits, 
or  the  increase  in  the  mechanical  energy.  Hence  equation  (53) 
immediately  follows. 

32.  The  Electrodynamic  Balance.  As  an  example  of  (51)  we 
shall  find  the  forcive  upon  a  circular  coil  (2)  of  n  turns  and  radius 
r  placed  with  its  center  in  the  axis  of  a  much  larger  circular  coil 
(i)  of  TV  turns  and  radius  R,  the  planes  of  the  coils  being  parallel. 
Let  the  currents  of  the  larger  and  smaller  coils  be  denoted  by  7T 
and  72  respectively,  and  the  distance  between  them  by  d.  Then 

Nu  =  pHnrr2n  =  TrrR^nNIJ  2(R2  +  d2)* 
M=  NJ^  =  7rriR2^nNl2(R2  +  d2)*  (54) 


H  being  practically  uniform  in  the  small  region  occupied  by  the 
small  coil,  and  /JL  (for  air)  being  sensibly  equal  to  unity.  The 
force  tending  to  increase  the  distance  d  is,  by  (51), 

F=  -  I^dMjdd  =  -  ^Trr-R^nNdl^l  2(R2  +  d2)*     (55) 


If  /j  and  /2  have  the  same  direction,  F  is  attractive,  otherwise 
repulsive.  If  the  same  current  /  is  caused  to  flow  through  the 
two  coils  in  series  in  the  same  direction,  the  force  is 

F=  -  ^7rr2R2^nNdI2l2(R2  +  ^2)i  (56) 

If  a  third  coil  C,  exactly  similar  to  the  larger  coil  A  is  placed 
with  its  plane  parallel  to  that  of  A  and  distant  therefrom  2df 
with  B  half  way  between  the  centers  of  A  and  C,  and  if  the  same 
current  /is  made  to  flow  through  A  and  C  in  opposite  directions, 
and  also  through  B,  the  force  F  will  be  twice  as  great  as  that 
given  by  (56);  or,  in  magnitude, 

d2)*  (57) 


364          ELEMENTS    OF    ELECTROMAGNETIC   THEORY 

If  the  coils  are  mounted  with  their  planes  horizontal,  and  if  the 
small  coil  B  is  connected  with  one  end  of  the  beam  of  a  balance, 
the  force  F  can  be  easily  measured,  and  /  determined  in  absolute 
measure.  We  have  in  this  case 

/  =  (R*  +  d^F^rR(^7riJidnN^  (58) 

Since  F  is  proportional  to  the  square  of  7,  alternating  as  well 
as  direct  currents  can  be  measured.  The  instrument  is  known 
as  an  electrodynamic  balance. 

For  descriptions  of  two  electrodynamic  balances  by  which  cur- 
rents have  been  determined  in  absolute  measure  with  great  pre- 
cision, see  Lord  Rayleigh,  Phil.  Trans.,  Part  II.,  1884,  and  H. 
Pellat,  Comptes  Rendus,  Vol.  103,  1886.  The  most  recent  of 
electrodynamic  balances,  that  of  von  Helmholz,  is  described  by 
Kahle,  Zeitschrift  fur  Instrumentenkunde,  Vol.  17,  1897. 

33,  The  Torsion  Electrodynamometer.  If  the  planes  of  the 
coils  A  and  B  of  the  last  article  make  with  one  another  an 

angle  0, 

M  =  7rr*R2nN  p,  cos  0/2(R2  +  d^  (59) 

and  there  is,  in  addition  to  the  force 

•F  =  -  IJjtMjdO  =  i7rr*R2nNd  ^//2  cos  0/2(R2  +  d^  (60) 
a  torque  in  the  direction  of  increase  of  the  angle  0  equal  to 

T=  -  I^dMjde  =  irr*B*nNIJjL  sin  0/2(R2  -f  d^    (61) 


If  the  smallest  coil  is  at  the  center  of  the  two  coils  A  and  C, 
§  32,  we  have 


and 

T^-I^dMjdO 

7r^R2nNp  cos  0/(R2  +  </2)*]  (62) 

sin  &/(R2 


ELECTROMAGNETIC    INDUCTION.  365 

from  which  the  equations  for  the  Weber  and  Siemens  forms 
of  electrodynamometer  immediately  follow,  as  in  §  34,  Chap- 
ter XII. 

For  a  description  of  one  of  the  most  recent  and  accurate  of 
torsion  electrodynamometers  see  Patterson  and  Guthe,  Physical 
Review,  Vol.  7,  1898. 

34.  General  Definitions  of  B  and  <E>.     In  XI.  ^  was  defined  by 
(2),  and  B  by  (5).     These  definitions  are  not  valid,  however,  in 
their  unmodified  form  without  qualification  when  the  magneti- 
sation is  wholly  or  partially  intrinsic  (§  22,  XL).     We  shall  now 
redefine  B  by  (9).     Thus 

B=-Tcur\Edt  (63) 

JQ 

the  substance  considered  being  in  its  neutral  state  at  the  time 
/=  o. 

Consistently  with  the  preceding  definition  of  B  and  XL,  we 
shall  redefine  3>  by  the  equation 

3>  =  f#  cos  OdS  =  -  f  "^dt  (64) 

the  substance  being  in  its  neutral  state  at  'the  time  t  =  o. 

<l>  and  B,  defined  by  these  equations,  can  evidently  be  deter- 
mined experimentally  (by  the  ballistic  methods  referred  to  be- 
low). Starting  with  these  definitions  it  can  be  shown  by  experi- 
ment that  tubes  of  magnetic  induction  are  always  closed,  as 
assumed  in  §  14,  XL  These  definitions  are  perfectly  consistent 
with  the  earlier  definitions,  and  are  more  general,  including  all 
cases  of  intrinsic  as  well  as  elastic  magnetisation. 

35.  Magnetisation    Curve.     Redefinition  of  /*.     Permeability. 

If  we  start  with  a  substance  in  a  neutral  state  and  increase  the 
magnetic  intensity  H  from  zero  in  a  series  of  steps,  observing 
the  corresponding  values  of  B,  we  obtain,  on  platting  the  results 
graphically,  a  curve  showing  the  relation  between  B  and  H  and 
called  the  magnetisation  curve  of  the  substance. 


366          ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

We  shall  now  redefine  /*,  for  any  given  value  of  H,  as  the  ratio 
of  B  to  H  for  the  given  point  on  the  magnetisation  curve.  The 
relation  B  =  pH  thus  holds  for  the  process  of  magnetisation  rep- 
resented by  the  curve. 

The  curve  showing  the  relation  between  J  =  B(  =  /x//)  —  ^H 
and  H  is  also  called  the  magnetisation  curve  of  the  substance. 
Either  curve  can  of  course  be  readily  obtained  from  the  other. 

The  permeability  of  a  substance  is  the  ratio  of  its  inductivity  to 
the  inductivity  of  the  standard  medium.  If,  as  in  this  book,  free 
aether  is  chosen  as  the  standard  medium,  the  permeability  p/fJ>Q 
of  a  substance  is  numerically  equal  to  its  inductivity.  The  in- 
ductivity of  air  is  only  slightly  greater  than  unity,  being  equal,  at 
ordinary  temperatures  and  pressures,  to  about  /XQ(I  -f-  3  x  io~6). 

36.  Diamagnetic  and  Paramagnetic  Substances,  The  inductivity 
fjL  of  nearly  every  substance  is  independent  of  the  value  of  H  and 
is  very  nearly  equal  to  /*0  =  I  -  Thus  the  magnetisation  curve 
of  such  a  substance  is  a  straight  line,  and,  if  B  and  H  are  platted 
to  the  same  scale,  makes  an  angle  of  very  nearly  45°  with  the 
axis  of  H. 

A  substance  whose  inductivity  is  less  than  JJL  or  whose  per- 
meability is  less  than  I ,  is  called  a  diamagnetic  substance  ;  and 
a  substance  whose  inductivity  is  greater  than  /AO  is  called  a  mag- 
netic, a  paramagnetic,  or,  if  its  inductivity  is  great  and  its  mag- 
netic properties  resemble  those  of  iron,  a  ferromagnetic  sub- 
stance. The  inductivity  of  every  diamagnetic  substance  is  very 
nearly  equal  to  unity,  water  being  the  commonest  example. 
The  inductivity  of  a  diamagnetic  or  weakly  magnetic  substance 
is  best  investigated  by  methods  analogous  to  those  of  §§  3  and 
4,  VII.  (see  A.  P.  Wills,  Physical  Review,  6,  1898;  Jager  u. 
Meyer,  Wied.  Ann.,  67,  1900;  Du  Bois,  The  Magnetic  Circuit). 
In  §§37  and  38  are  described  two  methods  commonly  applied  to 
iron,  nickel,  and  cobalt,  in  which  IJL  reaches  great  magnitudes. 

For  detailed  information  on  the  magnetic  properties  of  iron 
and  other  ferromagnetic  substances,  together  with  the  methods 


ELECTROMAGNETIC   INDUCTION.  367 

of  experimental  investigation,  reference  may  be  made  to  Ewing's 
Magnetic  Induction  in  Iron  and  Other  Metals  and  to  Du  Bois's 
The  Magnetic  Circuit.  The  more  important  magnetic  properties 
of  iron  are  briefly  described  in  Ch.  Maurain's  recent  Le  Magnet- 
isme  du  Per.  A  resume  of  the  magnetic  properties  of  matter 
generally,  with  abundant  references,  is  given  by  Du  Bois  in  Vol. 
II.  of  the  Rapports  of  the  International  Congress  of  Physics,  1900. 

37.  The  Determination  of  B,  /*,  and  J  (Intensity  of  Magnetisa- 
tion) by  the  Magnetometric  Method.  A  very  long  solenoid, 
similar  to  that  of  §  20  B,  XII.,  is  mounted  with  its  axis  vertical, 
and  a  magnetometer  (§  26,  XL)  is  mounted  with  its  needle  in 
vertical  plane  passing  through  the  solenoid  perpendicular  to  the 
magnetic  meridian  and  in  a  horizontal  plane  passing  through  the 
upper  part  of  the  solenoid.  The  solenoid  is  connected  in  series 
with  another  coil,  called  a  compensating  coil,  so  arranged  that 
both  coils  together  produce  no  deflection  of  the  needle  when 
traversed  by  a  current. 

The  iron  or  other  substance  whose  magnetisation  is  to  be 
investigated,  in  the  form  of  a  long  cylindrical  rod  similar  to  that 
of  §  20  B,  XII.,  is  then  placed  vertically  in  the  solenoid  traversed 
by  a  steady  current  with  its  axis  in  the  vertical  plane  perpen- 
dicular to  the  meridian  passing  through  the  needle,  and  its 
height  is  adjusted  until  a  position  is  reached  in  which  the  deflection 
of  the  needle  is  a  maximum.  In  this  position  the  upper  resultant 
pole  of  the  rod  is  approximately  in  the  horizontal  plane  passing 
through  the  magnetometer  needle.  Let  \L  denote  the  distance 
from  this  plane  to  the  center  of  the  rod.  Then  the  other  result- 
ant pole  is,  by  symmetry,  distant  approximately  \L  from  the 
center  on  its  other  side.  Let  R  denote  the  distance  from  the 
center  of  the  magnetometer  to  the  axis  of  the  rod. 

The  circuit  is  broken,  and  the  rod  is  demagnetised  (if  this  is 
necessary  for  the  purpose  in  view)  and  left,  or  reinserted,  in  the 
position  already  determined  for  maximum  deflection.  The  demag- 
netisation can  be  accomplished  by  heating  the  rod  to  redness,  or 


368          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

by  sending  a  current,  gradually  diminished  to  zero  while  its 
direction  is  rapidly  reversed,  through  the  solenoid  with  the  rod 
in  place,  or  by  other  means. 

The  circuit  is  now  closed  and  the  current  increased  to  produce 
the  desired  value  of  H  =  nl  (§  20  B,  XII.),  thus  developing 
poles  of  strengths  -f  m  and  —  m  in  the  rod,  and  the  quantities 
under  investigation  are  determined  as  follows. 

The  magnetic  intensity  at  the  magnetometer  needle  due  to  the 
two  poles  of  the  rod  is 


perpendicular  to  the  meridian. 

If  H  denotes  the  horizontal  component  of  the  earth's  mag- 
netic field  at  the  needle,  the  needle  will  be  deflected  through  an 
angle  0  such  that 

H=  H  tan  (9 

By  measuring  6,  H,  L,  R,  and  S,  n,  and  the  current  /,  the 
quantities  B,  J  =  B  —  y^//,  JJL  —  ^  can  then  be  calculated  from 
the  above  equations  and  those  of  §  20  B,  XII. 

If  the  experiments  are  not  performed  at  the  earth's  magnetic 
equator,  where  the  total  intensity  is  horizontal,  the  intensity 
parallel  to  the  rod  will  not  be  given  completely  by  H  —  nl  on 
account  of  the  vertical  component  of  the  earth's  intensity.  This 
component  can  be  neutralised  by  winding  another  coil  in  the 
solenoid  and  passing  through  it  a  suitable  current. 

If  after  adjusting  the  apparatus  we  start  with  the  current  zero 
and  the  rod  in  a  neutral  state,  and  then  increase  the  current  by 
steps,  observing  both  currents  and  corresponding  deflections,  we 
obtain,  on  platting  the  results  graphically,  the  magnetisation 
curve  of  the  substance.  The  magnetisation  curve,  and  the  curve 
showing  the  relation  between  /A  =  B  /  H  and  H  obtained  there- 
from, are  shown  in  Fig.  113  for  a  particular  sample  of  wrought  iron. 

When  great  accuracy  is  essential,  an  ellipsoid,  instead  of  a 
cylinder,  of  the  substance  under  investigation  must  be  used. 
See  E  wing's  treatise  above  referred  to. 


ELECTROMAGNETIC    INDUCTION. 


369 


38.  The  Determination  of  B,  /JL,  etc.,  by  the  Ballistic  Method 
(Rowland's  Form).  The  substance  to  be  investigated,  in  the  form 
of  a  rectangular  tore,  §23,  whose  thickness  R2  —  Rl  =  a  is  small 
in  comparison  with  Rv  is  wound  with  a  toroidal  coil  of  n  turns. 

This  coil  is  connected  in  circuit  through  a  reversing  key  with 
a  battery  of  constant  e.m.f,  a  current  meter,  and  a  rheostat 
whose  resistance  can  be  suddenly  varied. 


All  the  quantities  a  e  exp 
'  n  E.M.  units) 


Magnetic  Intensity 
Fig.  113. 

A  secondary  coil  of  n1  turns  is  wound  around  the  toroidal  coil 
and  connected  in  circuit  with  the  coil  of  a  ballistic  galvanometer. 
Let  the  resistance  of  this  secondary  circuit  be  denoted  by  R. 

The  relation  between  the  deflection  of  the  galvanometer  and 
the  charge  q  circulated  through  its  coils  is  supposed  known  from 
direct  experiment. 

Since  R2  —  R^  =  a  is  small  in  comparison  with  Rlt  the  mag- 
netic intensity,  given  accurately  by  (39),  is  nearly  constant 
throughout  the  tore  and  sensibly  equal  to 


H 


+ 


370         ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

The  magnetic  flux  across  every  section  of  the  tore  is  Bab,  and 
the  coil  flux  through  the  secondary  is  n1  Bab. 

If,  by  altering  the  resistance  of  the  circuit,  the  current  is  sud- 
denly changed  from  a  value  /0  to  a  value  /,  the  magnetic  inten- 
sity will  be  increased  by 


R^  (65) 

and  the  magnetic  induction  will  be  increased  by 

(66) 


where  q  is  the  charge  sent  through  the  galvanometer  circuit 
when  the  coil  flux  through  the  secondary  changes  from  n'BQab 
to  nf  Bab  (§  9),  and  is  known  from  the  observed  galvanometer 
throw. 

All  the  quantities  in  the  second  members  of  (65)  and  (66) 
being  determined  by  experiment,  H  —  HQ  and  B  —  BQ  are  known. 

Starting  with  any  value  of  H(V  and  the  corresponding  value 
of  BQ,  and  increasing  or  decreasing  the  current  suddenly  in  a 
series  of  steps,  current  and  galvanometer  throw  being  read  at 
each  step,  the  relation  between  B—  BQ  and  H—  HQ  can  thus 
be  obtained  for  as  wide  a  range  of  H  —  HQ  as  desired. 

If  we  start  with  the  current  zero,  and  the  substance  under  in- 
vestigation in  a  neutral  state,  //"Oand  BQ  are  zero,  and  if  we  increase 
H  in  a  series  of  steps,  we  get  the  magnetisation  curve  of  the 
substance.  From  corresponding  values  of  B  and  //  on  this 
curve  //.  can  be  found  by  division. 

For  additional  ballistic  and  other  methods,  see  the  references 
given  above.  See  Du  Bois,  Zeitschrift  fur  Instrumentenkunde  ', 
Vol.  20,  1900,  for  a  description  of  his  latest  magnetic  balance 
and  its  theory. 

39.  Magnetic  Hysteresis.  If  a  sample  of  iron,  nickel,  or 
cobalt  is  carried  repeatedly  through  a  given  magnetising  cycle, 
the  intensity  being  increased  to  GB,  Fig.  1  1  4,  then  reversed  to 
HE  =  —  GB,  then  reversed  to  GB,  then  reversed  again  to  HE, 


ELECTROMAGNETIC    INDUCTION. 


371 


and  so  on  for  a  number  of  cycles,  and  the  relation  between  B  and 
H  then  investigated  for  a  complete  cycle  by  the  magnetometric 
or  ballistic  method,  a  closed  symmetrical  curve  with  general 
characteristics  similar  to  those  of  the  figure  will  result.  The 
arrow  heads  indicate  the  direction  in  which  the  cycle  is  traversed. 
Thus  the  magnetisation  is  in  part  intrinsic.  The  induction 
OC  or  OF  is  called  the  residual  or  remanent  induction,  and  is  the 
maximum  value  of  the  intrinsic  induction.  The  ratio  of  the 
remanent  induction  0 C  or  OF  to  the  maximum  induction  OG  or 


HYSTERESIS  OF  ANNEALED 
PIANOFORTE  STEEL 


40  60 

Magnetic  lntensity( 
in  E..M.  units 


'Fig.   114. 

OH  is  called  the  retentiveness  of  the  substance  for  the  given 
cycle.  The  reversed  intensity  OD  or  OA  necessary  to  reduce 
the  intrinsic  induction  to  zero  is  called  the  coercive  force  or  coer- 
cive intensity  of  the  substance  for  the  given  cycle. 

The  closed  curve  is  called  a  hysteresis  cycle  since,  as  it  is  de- 
scribed, the  induction  always  lags  behind  the  intensity. 

The  area  of  the  curve,  viz.^HdB  from  any  point  such  as  B 
around  the  cycle  once  to  the  same  point  again,  represents  the 
work  per  unit  volume  done  in  carrying  the  substance  from  the 


372          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

state  denoted  by  B  (or  the  state  denoted  by  any  other  point  on 
the  curve)  through  the  complete  magnetising  and  demagnetising 
processes  indicated  by  the  cycle  to  the  same  state  again.  Hence 
§HdB  for  the  complete  cycle,  or  the  area  of  the  cycle,  represents 
the  energy  dissipated  in  heat  per  unit  volume  per  cycle  by 
hysteresis. 

The  area  of  the  cycle  is  the  same  or  very  nearly  the  same  when 
the  cycle  is  traversed  very  slowly  and  when  it  is  traversed  very 
rapidly  (see  references  given  below  and  Comptes  Rendus,  April 
20,  1903).  Thus  the  phenomenon  of  magnetic  hysteresis  is  not 
due,  except  perhaps  to  a  slight  extent,  to  anything  akin  to  vis- 
cosity (Cf.  §  2,  VI.). 

For  references  to  the  literature  on  magnetic  hysteresis,  see 
Chapter  IV.  of  Maurain's  Le  Magnetisme  du  Per  and  the  resume 
by  Warburg  in  Vol.  II.  of  the  Rapports  presented  to  the  Inter- 
national Congress  of  Physics,  1900. 

If  we  assume  the  relation  B  =  pH  to  hold  for  the  hysteresis 
cycle,  as  it  holds,  by  definition  of  p,  for  the  magnetisation 
curve,  fi  goes  through  all  values  from  -f  oo  at  C  to  —  co  at  F  dur- 
ing the  cyclic  process.  By  introducing  the  intrinsic  intensity,  k, 
however,  always  acting  in  the  direction  of  the  induction,  and  by 
writing  H  for  the  vector  sum  of  h  and  the  field  intensity  //', 
which  we  have  hitherto  denoted  by  //,  we  may  so  define  h  that 
the  relation  B  =  pH  =  p(h  -f  H'}  (vector  sum)  holds  univer- 
sally, and  leads  to  no  impossible  values  of  ft  (cf.  §  4,  VI.). 

The  magnetic  phenomena  of  iron,  nickel,  cobalt,  etc.,  includ- 
ing the  trend  of  the  magnetisation  curves,  hysteresis,  the  relation 
of  the  magnetic  phenomena  to  temperature,  etc.,  have  been 
largely  explained  by  the  molecular  theory  developed  by  Weber, 
Maxwell,  and  Ewing.  For  an  extended  treatment  of  the  molec- 
ular theory  and  a  discussion  of  its  experimental  confirmation 
reference  must  be  made  to  Ewing' s  Magnetic  Induction  in  Iron 
and  Other  Metals,  Chapter  XI. 

40-44.  The  Current,  etc.,  in  an  Electrical  System  Containing, 
in  the  General  Case,  Resistance,  Inductance,  and  Capacity,  Immersed 


ELECTROMAGNETIC    INDUCTION. 


373 


in  a  Medium,  or  Media,  of  Constant  Inductivity  and  Permittivity, 

Let  a  condenser  AB,  Fig.  115,  of  capacity  5  be  connected  in 
series  with  a  conductor  CDF  whose  inductance  is  L  and  whose 
capacity  is  negligible  in  comparison  with  that  of  the  condenser, 
and  an  agent  with  an  intrinsic  e.m.f.  M/\  the  total  resistance  of 
the  circuit  being  R.  Let  the  e.m.f.  ^  be  reckoned  positive  when 
directed  around  the  circuit  in  the  direction  CDF.  The  agent 
containing  the  e.m.f.  is  supposed  to  be  capable  of  being  instan- 


R&L 


D 
Fig.   115. 

taneously  inserted  in  or  removed  from  the  circuit,  the  resistance 
being  kept  constant,  by  suitable  switches.  Let  the  time,  /,  be 
reckoned  from  the  instant  at  which  ¥  is  inserted  or  cut  out. 
Let  q  denote  the  charge  of  the  plate  A,  I  =  dq\dt  the  current  in 
the  conductor  in  the  direction  CDF,  Fthe  voltage  from  A  to  B, 
at  the  time  t ;  and  let  q^  70,  and  VQ  denote  the  initial  values  of 
q,  7,  and  V. 

At  the  time  t  the  electric  energy  of  the  system  is 


the  electrokinetic  energy  is 

T=  \LP  = 
the  rate  of  dissipation  of  energy  in  heat  is 


374 


ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 


and  the  rate  at  which  energy  is  supplied  to  the  system  by  the 
agent  with  the  intrinsic  e.m.f.  is 


41.  Non-Inductive  Circuit.  Case  I.  Let  "SP  =  constant,  L  =  o, 
sensibly,  and  therefore  T~o. 

A.  Let  M*  be  suddenly  cut  out  of  the  circuit,  the  initial  charge 
of  A,  qQ,  being  equal  to  S"V.  By  the  principle  of  the  conversa- 
tion of  energy,  we  have  at  the  time  / 


or 


(66) 


a  relation  which  might  have  been  written  down  at  once  from 
Ohm's  law  \_dq\dt  =  /=  (VB  -  VA)fR  =  -  V\R  =  - 


0.00001 


0.00002  0.00003  0.00004 

Time  in  Seconds 


.  116. 


The   solution   of  (66),  with   the    condition   q  =  qQ 

/  =  o,  is 

q 
whence 


and 


/  =  dqjdt  =  -  ^IR  •  e-l'SR'<  =  IQe-l/SR'f 


when 

(67) 
(68) 
(69) 


ELECTROMAGNETIC    INDUCTION.  375 

The  relation  between  q,  /,  and  t  is  shown  graphically  for  a 
particular  case  in  Fig.  1 16,  Curve  I. 

The  time  SR  in  which  qjqQ  —  VjVQ=  IjIQ  becomes  i/e  is 
called  the  time  constant  of  the  system. 

B.   Let  M^  be  suddenly  inserted  into  the  circuit.     In  this  case 

By  the  principle  of  the  conservation  of  energy 
•Vdqldt** 


or 

"¥  =  qjS  -f  R  dqjdt  (70) 

another   equation    which    can    be    obtained    immediately   from 
Ohm's  law. 

To  solve  (70),  put  q  —  S^f  =  q' ,  and  the  equation  becomes 

Rdq'ldt+  q'lS=  O 
the  solution  of  which,  with  the  condition  q  =  o  when  t  =  o,  is 


whence 


^-i/^-')  (72) 

r™*-'  (73) 

The  relations  between  ^,  /,  and  t  are  shown  in  Fig.  116, 
Curve  II.,  for  the  same  system  whose  discharge  is  illustrated  in 
Curve  I. 

42.  Case  II.  Inductive  Circuit  Without  Capacity.    Let  ¥  =  con- 

stant, 5  =  infinity  (condenser  short-circuited),  and  therefore  W=.  o. 
A.   Let  ^  be  suddenly  cut  out  of  the  circuit,  the  initial  value 
of  the  current  being  /0  =  W/R.     In  this  case  we  have,  by  the 
principle  of  the  conservation  of  energy, 

-  d($LP}ldt  =  RP 
or 


3/6          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 


e/-o  (74) 

which    also    follows    immediately    from    Ohm's    law    [/=  (— 


The  solution  of  this  equation,  with  the  condition  /0  =  W/R,  is 
I  =  I  e~R/L't  =  "ty  I R' e~R/L't  (7$} 

The  relation  between  /  and  t  is  shown  for  a  particular  system 
in  Fig.  1 17,  Scale  A. 

The  time  L/R  in  which  /  falls  to  i/e  of  its  initial  value  is 
called  the  time  constant  of  the  circuit. 


1.0 


0.8 


|°-4 

3 

O 

0.2 


R  =10  Ohms 
L—  0.1  Henry 
V—  10  Volts 
-i.—  0.01  Secor 


-0.4-  ,5 

E 


0.01 


0.02 
Time  in  Seconds 

Fig.   117. 


The  total  electric  discharge  in  the  positive  direction  around 
the  circuit  after  the  time  t  =  o  is 

q  =  f  Idt  =  ^flR  f  e-R/L-*dt  =  DVJR*  =  LIJR     (76) 

Jo  Jo 

The  same  result  follows  from  (12): 

=  LIJR  =  LV/R2 

B.  Let  the  agent  with  e.m.f.  ¥  be  suddenly  inserted  into  the 
circuit,  the  initial  value  of  the  current  being  /0  =  o.  In  this  case 
the  principle  of  the  conservation  of  energy  gives 


ELECTROMAGNETIC    INDUCTION.  377 


or 

V^RI+Ldlfdt  (78) 

which  may  be   obtained  from   Ohm's  law  directly    [/=(¥•— 
LdHdi)\R~\. 

The  solution  of  this  equation  is 

!  -«"»*••)  (79) 


Thus  /  may  be  regarded  as  the  sum  of  a  steady  current  ^  JR 
and  an  induced  current 


The  relation  between  /  and  /  for  the  same  system  illustrated 
in  Case  II.  is  shown  in  Fig.  1  17,  Scale  B. 

The  total  electric  charge  traversing  the  coil  in  the  positive 
direction  due  to  the  induced  current  is 

q  =  -  LV/R2  (So) 

43.  Case  III.    Circuit  Containing  Both  Capacity  and  Inductance. 

Let  M/1  be  cut  out  of  the  circuit  at  the  time  t  =  o.     In  this  case 
the  principle  of  the  conservation  of  energy  gives 


-R(dqldff 
or 

Ld2qjdt2  +  Rdqjdt  +  qjS  =  O  (8 1) 

which,  like  equations  (66),  (78),  etc.,  also  follows  immediately 
from  Ohm's  law. 

To  solve  (81),  assume  q  =  constant  x  emt  and  substitute  in  the 
equation.     Thus  we  obtain 


i/S=o  (82) 

The  values  of  m  which  satisfy  this  equation  are 

ml=-R/2L  +  (R2!^  _  i  /SZ)*  =  -  a  +  (a*  -  P)* 
and 


ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

=  _  a  -  (a2  -  t 


if  we  put  R/2L  =  a  and  I  /SL  =  £2. 

If  m2  is  not  equal  to  mv  the  general  solution  of  (81)  is  there- 
fore 

q  =  Alf*  +  Af  (83) 

in  which  Al  and  A2  are  constants  to  be  determined  by  the  initial 
conditions  of  the  problem. 

When  m2  =  ml  —  m  =  —  a,  the  solution  of  (81)  is 

q  =  (pi  +  Bfr-«  (84) 

where  Bl  and  B^  are  constants  to  be  determined  by  the  initial 
conditions. 

Three  cases  are  to  be  considered  :  A,  when  a2  =  IP ;  B,  when 


0,000005  0.000010  0.000015  0.00002 

Time  in  Seconds 


A.  a2  =  &*.     Since  qQ 

2     ~~~    1  0\ 


Fig.  118. 

and  70  =  o,  (84)  gives 


and 


7=  - 


(85) 
(86) 
(87) 


The  relations  between  ^,  7,  and  the  time  for  a  particular  system 
are  shown  in  Fig.  118,  I.  and  II.  (Scale  A). 


ELECTROMAGNETIC    INDUCTION.  379 

B.  #2  >  &.     To  determine  the  constants  Al  and  A2  we  have, 
when  t  =  o, 


and 

/o  =  O  =  /tf^j  4- 

Hence  (83)  becomes 


from  which  we  obtain 

(89) 


and 

/  =  dqfdt  =  S^m^mJ^  -  m^  -  (f*  -  *"*)  (90) 


The  curves  showing  the  relation  between  q  and  t  and  the 
relation  between  7  and  /  are  very  similar  to  the  corresponding 
curves  of  Fig.  118.  Their  drawing  is  left  to  the  reader. 

C.  Oscillatory  Discharge,  a2  <  #*.  Equation  (83)  may  be 
written 

q  =  e-^A/W-^'  +A2e-w2-a^t) 

where  i  denotes  (—  i)*.     This  equation  is  equivalent  to 

q  =  e~at\A  cos  (^  -  a^t  +  B  sin  (P  -  a2)*f\  (91) 

where  A  and  B  are  real  constants  to  be  determined  by  the  initial 
conditions.     From  (91) 

/=  dq\dt  =  e~at{  [^(^2  -  a2)^  -  Ad\  cos  (^  -  a^t  -  (     . 


a*)*  +  Ba\  sin  (b2  -  az^t] 
From  the  initial  conditions  q  =  qQ  —  SW,  and  /=  70  =  o,  when 


t  =  o,  the  above  equations  may  be  written 

-  ^2)^  •  sin  (P  -  a^t]  (93) 


and  .  ; 

7  =  -  SVP/(P  -  aj  -  e~at  sin  (<P  -  ^2)*/  (94) 

We  have  also 


380 


ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 


The  relations  between  qy  /,  and  t  are  shown  for  a  particular 
case  in  Fig.  1 19.     (See  the  data  given  in  the  figure.) 


0.1  OOi 


-10* 


O.OOC5  0.0010 

Time  in  Seconds  and  in  Periods  of  Oscillation 

Fig.  119. 


0.0015 


The  discharge  is  seen  to  be  oscillatory,  as  well  as  damped,  the 
charge  of  the  condenser  and  the  current  in  the  wire  each  revers- 
ing its  sign  at  intervals  of  \Tt  where  T  is  the  period  of  the  oscil- 
lation and  is  given  by  the  equation 


27T/T=  (tf  — 


or 


(95) 


and   the   amplitude  of  the  oscillation   being  gradually  reduced 
to  zero. 

The  charge  q  is  zero  at  times  /  such  that 

/  =  T/27T  •  \TT  -  tan-1^2  -  a2)1*  /a] 
and  /  is  zero  at  times  t'  such  that 

t'  =  \Tn 

where  n  is  zero  or  any  integer. 

Each  quantity  reaches  a  maximum  or  minimum  value  half  way 
between  two  successive  zero  values. 


ELECTROMAGNETIC    INDUCTION.  381 

The  current  reaches  a  maximum,  minimum,  or  zero  value  ahead 
of  the  charge  by  the  interval 


T  ( 

(  TT  - 
27r  \ 


tal1"  -         M- 


6/27T-  T  is  called  the  phase  difference  between  the  current  and  the 
charge,  and  0  the  angle  of  lag  of  the  charge  behind  the  current, 
or  the  angle  of  lead  of  the  current  over  the  charge. 

The  ratio  of  the  magnitude  of  a  maximum  or  minimum  value 
of  the  current  or  charge  to  the  magnitude  of  the  next  following 
minimum  or  maximum,  occurring  \T  later,  is 


The    natural   logarithm    of  this   ratio  is   called   the  logarithmic 
decrement  of  the  oscillation  and  is  denoted  by  X.     Thus 


i)*  (96) 

If  R  =  o,  a  =  o,  no  energy  is  dissipated,  and  the  oscillation 
takes  place  without  damping  (energy  radiated  being  assumed 
zero).  The  period  when  R  =  o  is 

ro=2,r(SZ)*  (97) 

By  (95),  (96),  and  (97)  we  have 

T=  Tt(i  +  X'/**)*  =  TJi  +  JXY**  +  •  •  -)  (98) 

Hence  the  effect  of  a  small  decrement  on  the  period  is  propor- 
tional to  the  square  of  the  decrement.  See  Fig.  1  19  for  the  rela- 
tion between  T  and  TQ  for  the  circuit  whose  discharge  is  there 
illustrated. 

If  R  is  increased  while  L  and  5  remain  constant,  the  period 
of  the  oscillation,  as  well  as  the  damping,  is  increased  until,  when 
a"  =  b2,  the  oscillatory  character  of  the  discharge  disappears.  As 
the  resistance  is  still  further  increased,  the  discharge  assumes 
more  and  more  nearly  the  character  of  the  discharge  of  a  con- 
denser through  a  non-inductive  resistance  (Case  L,  A). 


382  ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

The  maximum  and  minimum  ordinates '  of  the  current  curve 
are  equal  to  the  corresponding  (to  the  same  time)  ordinates  of 
the  logarithmic  curves  -f  S^fe~at  and  —  S^fe~at,  or  the  logarith- 
mic curves  are  tangential  to  the  current  curve  at  the  maximum 
and  minimum  points. 

The  maximum  and  minimum  ordinates  of  the  charge  curve 
exceed  the  corresponding  ordinates  of  the  logarithmic  curves, 
which  are  therefore  not  tangential  to  the  charge  curve.  The 
ratio  of  a  maximum  or  minimum  ordinate  of  the  charge  curve 
to  the  corresponding  ordinate  of  the  logarithmic  curves  is 
bl^  —  #2)*,  which  is  nearly  unity  when  ajb  is  small.  For  the 
case  illustrated  in  the  figure  this  ratio  is  about  1000  to  995,  a 
difference  scarcely  perceptible  in  the  drawing. 

.For  one  of  the  most  recent  and  accurate  experimental  investi- 
gations confirming  the  above  theory,  which  is  due  to  Lord  Kelvin 
(1853),  the  reader  is  referred  to  a  memoir  by  Webster  (Physical 
Review,  6,  p.  297,  1898),  where  references  to  the  earlier  literature 
will  be  found. 

44.  Case  IV.  Periodic  E.M.F.  Let  the  electromotive  force  be  a 
simple  harmonic  function  of  the  time,  M*  ="*PQ  cos  //,  where  /  = 
27r/T=  27rn,  T  being  the  time  of  one  complete  period,  and  n 
the  number  of  periods  per  second,  or  the  frequency,  of  the  elec- 
tromotive force. 

In  this  case  we  have,  by  either  of  the  methods  already  fre- 
quently employed, 

Ld*qjdt2  +  Rdqjdt  +  qfS  =  ^0  cos  //  (99) 

The  general  solution  of  this  equation  is  the  general  solution 
of  (81)  already  obtained,  viz.  (83),  -f  a  particular  solution  of 
(99).  To  obtain  a  particular  solution  of  (99),  assume 

q=A  cos  (pt—0) 

substitute  in  (99),  and  equate  to  zero  the  coefficients  of  sin  // 
and  cos  pt  separately  (since  the  resulting  equation  must  be  true 


ELECTROMAGNETIC    INDUCTION.  383 

for  all  values  of  /).     In  this  way  we  find  that  the  above  equation 
is  a  particular  solution  of  (99)  if 


(ioo) 
and 

B  =  ta 
Hence 


cos       t  - 


Since  the  last  two  terms  become  sensibly  zero  a  very  short  time 
after  closing  the  circuit,  we  may  write,  except  during  this  interval, 


q  =  T0  cos  (ft  -  e)lp\S?  +  p\L  -  I  /5/)2]*       (102) 

¥„  COS  (ft—0+  IT  1  2) 

'-^-r^+T^HTOT 
and 

V-qlS=-9,cos(pt-ff)ISp[.It?+f\L-  l/S/)2]*    (104) 

The  angle  0  is  called  the  phase  difference  (a  term  more  properly 
applied  to  6/27T-  T)  between  q  and  M',  or  the  angle  of  lag  of  ^ 
behind  "SP,  or  the  ##£•/,?  of  lead  of  M*  over  ^.  Similarly  6  —  JTT  =  0' 
is  the  angle  of  lag  of  /  behind  *P,  or  the  angle  of  lead  of  ^  over  /. 

The  quantity  [R2  +  /(Z  —  i/S/2)2]*  is  called  the  impedance 
of  the  circuit  (or  portion  of  the  circuit  considered),  the  quantity 
p(L  —  i  jSp2}  its  reactance,  and  the  quantity  I  /S/2  its  condensance. 

From  the  way  in  which  ,S  and  L  enter  into  the  above  equa- 
tions it  is  evident  that  a  condenser  of  capacity  5  in  series  in  the 
circuit  has  the  same  effect  upon  a  simple  harmonic  current  as 
would  a  negative  inductance  equal  to  I  /  '  Sp2.  Thus  condensance 
and  inductance  can  be  made  to  neutralise  one  another's  effects. 

For  given  values  of  ^  and  R  the  amplitude  of  /  reaches  a 
maximum  when  L  —  I  /  '  Sp2  —  o,  that  is  when  /  =  /0  =  27r/?"0, 
where  TQ  =  27r(LS)*  is  the  natural  period  in  which  the  system 
would  execute  oscillations  if  its  resistance  were  zero.  In  this 


3^4 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


case  6f  =  o,  the  current  is  in  phase  with  the  e.m.f.,  and  has  at 
all  times  the  same  value  as  if  there  were  no  inductance  or  con- 
densance  in  the  circuit.  In  this  case  the  circuit  is  said  to  be  in 
resonance  with  the  e.m.f.  The  relation  between  the  maximum 
value,  or  amplitude,/^,  of  the  current  and  TjTQ  for  a  given  value 
of  L/SR2  [see  equation  (100)]  is  shown  in  Fig.  120. 


i.o 


o.s 


0.2 


| 

L 
SR2= 

10 

/|\    RESjONANCE 
|    \    IN'DUCTAN 

OF  A  S 
CE  AND 

YSTEM 
CAPACI 

/VITH 

rv 

4M 

R 

•  1 

dp 

V 

/ 

i 

\ 

/ 

/ 

i 
i 
i 

^fc 

\-^ 

. 

-^-^"o 

^ 

2             0 

X 

4            0 

6            0 

i 
i 
i 

8            HO            1 

2             1 

4             1 

6             1 

8             2 

Ratio  of  T  to  To 
Fig.  120. 

Equation  (104)  may  be  written 

F=  F0  cos  (//—  0) 


where 


VQ  being  the  amplitude  of  V.  If  the  denominator  in  this  equa- 
tion is  less  than  unity,  which  is  easily  possible,  VQ  will  be  greater 
than  ^0,  the  amplitude  of  the  applied  e.m.f.  Differentiation  of 
(105)  shows  that  for  given  values  of  S,  L,  and  R  the  amplitude 
VQ  will  be  a  maximum  when 

/  =  (2L  -  SX2)/2SL2 

If  SR2  is  small  in  comparison  with  2L,  this  expression  gives, 
approximately,  pz  =  i/SL,  or 


ELECTROMAGNETIC    INDUCTION.  383 

r=r0  =  2^(5Z)i 

the  natural  period  of  the  system  without  damping.      In  this  case 


which,  in  accordance  with  the  above  assumption  (2L/R2S  large), 
is  much  greater  than  unity.  This  is  another  illustration  of  the 
effects  of  resonance. 

Equation  (103)  may  be  written 

7=/0  cos  (//-*') 

The  power  supplied  to  the  circuit  by  the  e.m.f.  ¥  at  the  time  t  is 
/>=¥/=  ^0/0  cos  (//  -  0')  cos  pt  (i  06) 

It  is  easy  to  show,  either  graphically  or  by  means  of  (106), 
that  when  6f  =  o,  P  is  always  positive  ;  that  when  Of  is  equal  to 
d=  7T/2  (its  limiting  values  in  the  system  considered),  Pis  posi- 
tive during  half  the  period  and  negative  during  half  the  period, 
no  power  on  the  whole  being  developed  by  the  e.m.f.  (to  make 
6f  =  db  7T/2,  R  must  be  zero  and  either  the  inductance  or  the  con- 
densance  must  be  zero)  ;  and  that  when  6f  is  less  than  ±  Tr/2 
and  greater  than  zero,  P  is  positive  during  more  than  half  the 
period. 

45.  Dynamical  Analogues.  Consider  the  angular  motion  of  a 
cylinder  C  about  the  vertical  axis  of  a  suspending  wire  AB. 
Let  the  moment  of  inertia  of  C  about  this  axis  be  denoted  by  L, 
and  suppose  the  inertia  of  the  wire  negligible.  When  the  cylin- 
der is  turned  through  an  angle  q,  the  twist  of  the  wire  gives  rise 
to  a  return  torque  V=  qjS,  i/S  being  a  constant  of  the  wire 
depending  upon  its  rigidity  and  dimensions  and  called  its  elas- 
tance  or  the  reciprocal  of  its  permittance,  tending  to  diminish  q. 
Let  the  motion  of  the  cylinder  be  resisted  by  a  frictional  torque 
proportional  to  its  angular  velocity  dqjdt  =  7,  that  is  by  a 
torque  —  Rf=  —  Rdqjdt,  where  R  is  a  constant.  At  the  time 


386          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

t  =  o,  let  a  torque  "SP,  capable  of  producing  a  final  angular  dis- 
placement qQ  =  SW,  be  applied  to  the  system,  or  removed  after 
having  been  applied  sufficiently  long  to  produce  the  maximum 
displacement  qQ.  Then  the  equation  of  motion  of  the  cylinder 
after  ^  has  been  applied  or  removed  may  be  found  as  follows  : 
The  potential  energy  of  the  system  is 


the  kinetic  energy  of  the  system  is 


the  rate  of  dissipation  of  energy  in  heat  is 

F=RP=  R(dqjdtf 
the  rate  at  which  energy  is  supplied  to  the  system  is 


By  the  principle  of  the  conservation  of  energy 

¥/  =  dWjdt  +  dTjdt  +  F 
whence 

*  -f  Rdqjdt 


which  is  the  equation  of  motion  sought. 

The  equation  may  be  obtained  also  by  the  direct  application 
of  the  second  law  of  motion.  Thus,  ^P,  the  total  torque  acting 
upon  the  cylinder,  consists  of  three  parts  :  A  torque  V=  qjS  to 
balance  the  counter  torque  due  to  the  torsion  of  the  wire,  a 
torque  Rl  =  Rdqjdt  to  balance  the  torque  —  RI  due  to  friction, 
and  a  torque  Ldljdt  =  Ld^qjdt*  to  balance  the  torque  —  Ldljdt 
due  to  the  inertia  of  the  cylinder,  or  to  produce  the  angular 
acceleration  dl\dt.  Hence 

¥  =  Ld*qjdt*  +  Rdqjdt  -f  qjS 
The  equation  of  motion  after  the  removal  of  is  ^ 
Ld^qjdt*  +  Rdqjdt  -f  qjS  =  o 


ELECTROMAGNETIC    INDUCTION.  387 

In  the  same  way,  if  a  periodic  torque  M*  =  "¥0  cos  pt  acts  upon 
the  cylinder,  the  equation  of  motion  is 

Ld*qjdt-  +  Rdqjdt  +  qjS  =  ^0  cos// 

The  solutions  of  these  equations  are  given  above. 

I.  Let  the  inertia  of  the  cylinder  be  negligible  (Tj  J/F"  sensibly 
zero).     A.  The  cylinder  is  displaced  initially  through  the  angle 
#0,  and  the  wire  possesses  the  potential  energy  \q* /S.     If  the 
cylinder  is  suddenly  released,  it  will  begin  to   move  with  the 
angular  velocity  /=  q^j SR,  which  gradually  diminishes  toward 
zero  as  <?Q  diminishes.      The  potential  energy  is  dissipated  by 
friction  during  the  process.     The  relation  between  q,  7,  and  /  is 
shown  in  Fig.  1 16,  I. 

B.  If  a  constant  torque  Mf  is  applied  to  the  cylinder  at  rest  in 
its  equilibrium  position,  it  will  suddenly  acquire  an  angular  ve- 
locity /=  *&  I R,  which  will  decrease  toward  zero,  owing  to  the 
return  torque  exerted  by  the  wire,  as  the  angular  displacement 
increases  toward  the  limiting  value  qQ=S^.  The  potential  energy 
will  increase  during  the  process  toward  the  limiting  value  -^q,2  J  S, 
and  an  equal  quantity  of  energy  will  be  dissipated  by  friction. 
The  relations  between  q,  7,  and  t  are  shown  in  Fig.  116,  II. 

II.  Let  the  elastance  of  the  wire  be  negligible,  or  let  the  wire 
be  removed  and  let  the  cylinder  be  supported  on  pivots  (WjT 
sensibly  zero).      A.   If  a  constant  torque  "SP  is  applied  to  the  cylin- 
der, its  velocity  and  kinetic  energy  will  increase  from  zero  toward 
the  limiting  values  f=V/R  and   T=\LP.     The  torque  will 
continue  to  dissipate  energy,  the  limiting  rate  being  "W2/R. 

B.  If  the  torque  is  suddenly  removed,  the  cylinder  will  con- 
tinue to  rotate  with  continually  diminishing  velocity  and  energy 
until  the  energy  is  wholly  dissipated  in  heat  by  friction. 

The  relations  between  /  and  t  for  the  two  cases  are  given  in 
Fig.  117. 

III.  Let  both  the  elastance  of  the  wire  and  the  inertia  of  the 
cylinder  be  noticeable  (W  and  T  both  appreciable).     A  and  B. 
Let  R2/4L2  be  equal  to  or  greater  than  ifSL.     If  the  cylinder, 


388  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

initially  displaced  though  an  angle  qQJ  the  potential  energy  being 
^q^jS,  is  suddenly  released,  the  potential  energy  will  decrease, 
the  velocity  and  kinetic  energy  will  increase,  reach  a  maximum, 
and  decrease,  with  q,  indefinitely  toward  zero,  all  the  energy  be- 
ing finally  converted  into  heat. 

If  the  torque  is  suddenly  applied  to  the  cylinder  at  rest  in  its 
equilibrium  position,  the  velocity  and  kinetic  energy  increase 
gradually  from  zero  to  a  maximum,  decrease,  and  approach  zero 
as  the  displacement  and  potential  energy  approach  their  limiting 
values  qQ  and  \q^fS.  During  the  process  an  amount  of  energy 
^q*jS  is  dissipated  in  heat. 

The  relations  between  q,  7,  and  t  for  the  first  case  are  shown 
in  Fig.  1  1  8. 

C.  Let  i/SL  be  greater  than  R2/4L2.  First  assume  R  to  be 
zero.  The  initial  displacement  qQ  and  potential  energy  \q2jS 
will  decrease  to  zero  as  the  velocity  /  and  the  kinetic  energy  T 
increase  to  maxima  when  the  cylinder  is  in  the  equilibrium  posi- 
tion. The  kinetic  energy  will  carry  the  cylinder  beyond  this  posi- 
tion until  the  displacement  is  —  ^0,  when  the  kinetic  energy  and 
velocity  will  be  zero  and  the  potential  energy  equal  to  its  initial 
value.  The  same  phenomena  will  then  recur  in  inverse  order, 
and  so  on  indefinitely,  the  time  of  a  complete  oscillation  being 


When  P^I^I?  is  not  zero,  the  phenomena  will  be  similar  ex- 
cept that  the  motion  will  be  damped,  each  elongation  being  less 
than  that  immediately  preceding,  since  energy  is  continually  dis- 
sipated by  friction.  The  time  of  an  oscillation  will  be  increased, 
and  the  time  between  zero  elongation  and  zero  velocity  will  not 
be  exactly  one  quarter  of  a  period. 

The  relations  between  q,  /,  and  /  are  shown  in  Fig.  1  1  9. 

IV.  If  an  alternating  torque  "SP  =  M^  cos  pt  is  applied  to  the 
cylinder,  it  will  make  simple  harmonic  vibrations,  after  the 
motion  has  become  steady,  with  the  period  T=  2irjp  of  the 
applied  e.m.f.  (99)  may  be  written 

q]S  =  T=  ¥0  cos  //  -  Ld-qjdt2  —  Rdqjdt 


ELECTROMAGNETIC   INDUCTION.  3^9 

If  Tj  TQ  is  very  great,  the  velocity  /  =  dqjdt,  the  acceleration 
d2qjdfi,  and  the  second  and  third  terms  of  the  second  member, 
which  are  the  counter  torques  of  inertia  and  friction,  are  very 
small,  so  that  V  =  q/S,  the  return  torque  of  the  twisted  wire,  is 
approximately  equal  to  M*,  the  applied  e.m.f,  and  its  maximum 
value  VQ  is  approximately  equal  to  ^0,  while  the  maximum 
elongation  is  approximately  equal  to  SWQ. 

If  T/T0  is  very  small,  or  ///0  very  great,  the  counter  torque 
of  inertia  —  Ld^qjdt*  is  great  ;  for  the  acceleration  d2qjd?  is 
proportional  to  p  and  to  the  amplitude  of  the  velocity  dqjdt  and 
is  therefore  great  even  if  this  amplitude  is  small.  Hence  the 
maximum  values  of  q  and  V,  the  twist  and  torque  of  the  wire, 
are  small.  The  velocity  dqjdt  is  also  small. 

If  T=  T0,  the  torque  of  inertia  is  just  balanced  by  the  return 
torque  of  the  wire,  or  the  first  member  of  the  above  equation 
and  the  second  term  of  the  second  member  cancel.  Hence  the 
velocity  reaches  its  maximum  value,  the  whole  torque  ^0  cos  pt 
being  applied  to  keeping  up  the  velocity  or  overcoming  friction. 
The  amplitude  of  the  velocity,  and  therewith  the  maximum 
elongation  of  the  cylinder  and  return  torque  of  the  wire,  increase 
until  the  rate  of  dissipation  of  energy  by  friction  is  equal  to  the 
rate  at  which  energy  is  supplied  by  the  e.m.f.  M*.  If  the  friction 
is  small,  the  maximum  elongation  and  return  torque  may  be 
much  greater  than  SW0  and  WQ,  which  would  be  produced  by  a 
steady  torque  WQ. 

46.  The  E.M.F.  Developed  by  Rotating  a  Coil  in  the  Earth's 
Magnetic  Field.  One  of  the  most  obvious  ways  of  developing  a 
simple  harmonic  e.m.f.  of  known  period  and  amplitude  is  to 
rotate  with  constant  angular  velocity  about  a  vertical  axis  in  the 
earth's  magnetic  field  a  coil  of  insulated  wire  wound  upon  a 
rigid  frame  with  the  planes  of  its  turns  parallel  and  vertical,  the 
ends  of  the  coil  being  connected  to  conducting  rings  revolving 
on  the  same  axis,  the  springs  bearing  on  these  rings,  connecting 
the  terminals  of  the  coil  with  the  rest  of  the  circuit. 


390 


ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 


Let  n  denote  the  number  of  turns  in  the  coil,  5  the  average 
area  enclosed  by  a  single  turn,  /  the  angular  velocity,  6  =  pt  the 
angle  made  by  the  planes  of  the  turns  with  the  magnetic 
meridian  at  the  time  t,  and  B  =  //.H  the  horizontal  component  of 
the  earth's  magnetic  induction  (sensibly  equal  in  magnitude  to 
H,  the  horizontal  component  of  the  intensity)  in  the  region 
occupied  by  the  coil. 

Then  at  the  time  t  the  coil  flux  due  to  the  earth's  field  is 

N=  nSB  sin  0 
and  the  intrinsic  e.m.f.  developed  in  the  coil  by  its  rotation  is 


=  _  dNjdt  =  —  nSB  cos  0  dOjdt 

=  —  pnSB  cos  //  =  WQ  cos  pt 


(107) 


For  an  accurate  determination  of  a  resistance  in  absolute 
measure  by  a  method  involving  essentially  the  use  of  such  a  coil 
with  known  constants,  see  Lord  Rayleigh,  Phil.  Trans.,  1882, 
Part  II. 

47,  A  Simple  Harmonic  E.M.F.  Acting  on  Two  Conductors  With- 
out Mutual  Inductance  Connected  in  Multiple.  If  an  e.m.f. 


S2 


Fig.  121. 


*P0  cos  pt  is  applied  to  the  terminals  of  the  conductors,  I  and  2,  it 
is  clear  from  §  44  that  the  currents  7t  and  /2  in  the  conductors 


ELECTROMAGNETIC    INDUCTION.  39  1 

will  be  simple  harmonic  with  the  period  T=  2Trjp  of  the  e.m.f., 
and  that  their  amplitudes  Al  and  A2  will  be  in  the  inverse  ratio  of 
the  impedances.  Thus 


Suppose  L2=  S1  =  Ri  =  R2  =  o,  approximately  (Fig.  121). 
Then 

AM=i//^A  (109) 

approximately. 

When  /  is  such  that  fSJL^  =  I,  or  T=  2ir(S2L^  =  T0,  the 
natural  period  in  which  the  system  12  would  execute  electrical 
oscillations  if  isolated  and  without  resistance,  etc.,  Al  =  A2  =  A. 
The  currents  in  the  two  branches  have  in  this  case  the  same 
direction  around  the  circuit  12,  and  the  e.m.f.  W  cos  pt  can  be 
removed  after  the  oscillations  are  started  without  affecting  the 
phenomena  (dissipation  and  radiation  of  energy  being  supposed 
zero).  That  /x  and  /2  are  in  this  case  opposite  when  measured 
from  A  to  B,  or  have  the  same  direction  around  the  circuit  1  2,  is 
clear  from  (100).  For  in  this  case  we  have 


0l  =  tan-!(  -RjpL^  =  TT  -  \axrlRllpLl  =  TT 
and  62  *=  tan~lpR2S2  =  o 

so  that 

/!  =  A  cos  (pt  —Ol  +  %ir)  =  A  cos  (pt  —  \ir] 

=  -  A  cos  (pt  +ITT)  =  -  A  cos  (pt  -  62  +  1  TT)  =  -  /2 

Since  the  current  in  the  external  circuit  connected  at  A  and  B 
is  7^/2=  o,  no  power  is  supplied  by  the  external  e.m.f.  This 
also  follows  from  the  consideration  that  /x  lags  90°  behind  ¥ 
and  72  is  in  the  lead  of  ^  by  90°. 

This  is  another  example  of  electrical  resonance. 

48.  Distribution  of  the  Total  Discharge  Through  Two  Coils  m 
Multiple.  Let  the  terminals  of  the  two  coils,  I  and  2,  be  joined 


392  ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

at  A  and  B,  Fig.  122,  and  let  the  e.m.f.  from  A  to  £  at  the  time 
t  be  denoted  by  ¥".  The  discharge  may  be  that  from  a  condenser, 
as  shown  in  the  figure,  or  any  other  form  of  discharge,  and  the 


Fig.  122. 

circuits  may  have  any  inductances  and  mutual  inductance  what- 
ever.    The  capacity  in  each  coil  is  supposed   negligible,  so  that 
the  current  is  the  same  across  every  section  of  each  conductor, 
no  charge  accumulating  at  any  point. 
The  total  charge  traversing  coil  i  is 


dt  =  i  IRV  dt 
and  the  total  charge  traversing  circuit  2  is 


^,  fv  and  I2  being  considered  positive  when  directed  from  A  to  B. 
Thus  the  total  discharge  q  =  gl  +  q2  is  divided  between  the 
two  coils  in  the  ratio 


that  is,  in  the  inverse  ratio  of  their  resistances.  Thus  the  total 
discharge  (considered  positive  when  in  one  direction,  negative 
when  in  the  other)  does  not  depend  upon  the  inductances,  but 
only  upon  the  resistances. 

49.  An  Electrical  System  (Transformer)  Consisting  of  Two  Cir- 
cuits, One  of  Which  Contains  an  Intrinsic  E.M.F.  M>.  We  shall 
consider  only  the  case  in  which  the  capacities  of  both  circuits  are 
negligible.  Let  the  resistances  and  inductances  of  the  two  cir- 


ELECTROMAGNETIC    INDUCTION.  393 

cuits  be  denoted  by  Rv  R2,  Lv  L2,  respectively,  and  their  mutual 
inductance  by  M\  and  let  the  time  be  reckoned  from  the  instant 
at  which  the  e.m.f.  is  applied  (to  circuit  i),  or  removed,  or  the 
resistance  of  circuit  I  suddenly  altered. 

Case  I.     "\P  =  Constant,     The  total  energy  of  the  system  is 
electro-kinetic  and  equal  to 


The  rate  at  which  energy  is  dissipated  in  heat  in  the  two  circuits  is 
F—  F  4-  F  —  R  72  -\-  R  f2 

fl  "I     •*•  2  —   7Vlyl       •     "*V272 

The  rate  at  which  power  is  supplied  to  the  two  circuits  is 


Therefore,  by  the  principle  of  the  conservation  of  energy, 


or 

I^dljdt  +  Mdljdt  +  R^  -  ¥) 

+  I&LJIJdt  +  Mdljdt  +  R2I2)  =  o 

We  proceed  to  show,  quite  independently  of  what  precedes, 
that  each  of  the  expressions  within  parentheses  in  this  equation 
vanishes  separately  ;  that  is,  that 

LJIJdt  +  Mdljdt  +  RJ,  -  ¥  =  o  (in) 

and 

-f  Mdljdt  +  R2f2  =  o  (112) 


equations  from  which  II  and  72  are  determined  below. 

The  validity  of  these  equations  can  be  established  in  several 
ways.  Thus  the  rate  at  which  electromagnetic  energy  is  gener- 
ated in  circuit  I  plus  the  rate  at  which  energy  is  transferred  from 
circuit  2  to  circuit  I  is  (9  —  Mdljdt}!^  while  the  rate  at  which 
the  electromagnetic  energy  of  circuit  I  increases  plus  the  rate  at 
which  energy  is  dissipated  by  its  resistance  is  d^L^I^jdt  +  R^- 
Hence 


394         ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 
(¥  -  Mdljdt)!, 


from  which  (i  1  1)  results  immediately. 

(111)  also  follows  immediately  from  Ohm's  law,  the  impressed 
e.m.f.  in  the  circuit  being  M*  —  L^dl^dt  —  Mdljdt. 

(112)  is  obtained  in  exactly  the  same  way  as  (i  1  1),  the  intrin- 
sic e.m.f.  in  circuit  2  being  put  equal  to  zero. 

If  we  place 

/,  -  */*,  =  // 


(in)  and  (i  12)  may  be  written 

L^jdt  +  Mdljdt  +  RJ1  =  o  (a) 

and 

L2dIJdt  +  Mdl^jdt  +  R2I2  =  o  (b) 

To  solve  these  equations,  put 

//  =  Af"  and  I2  =  Bemt  (c) 

where  A,  B,  and  m  are  constants  to  be  determined,  and  substi- 
tute in  (a)  and  (£).  On  dividing  the  resulting  equations  by  emt, 
we  obtain,  as  the  conditions  which  the  constants  must  satisfy  in 
order  that  (c)  may  give  the  solution, 

(Ljm  +  R^)A  +  MmB  =  o  (d) 

MmA  +  (L2m  +  R^B  =  o  (e) 

Eliminating  A  and  B,  we  obtain,  to  determine  m,  the  equation 

(L,L,  -  M*)m>  +  (R2L,  +  R.L^m  +  X&  =  o 
whose  roots  are 

-  (RJL,  +  RJj  +  [(/?/,  -  RJJ  +  4JW/1]*  ,    , 


and 

-  -  - 


ELECTROMAGNETIC   INDUCTION.  395 


Since  (RJ^  -  R^Ltf  =  (RJ^  -f  R^Ltf  —  4RlR2L1Lh  and  since 
L^L2  —  M2  =  o,  the  quantity  under  the  radical  is  positive  and 
numerically  less  than,  or  equal  to,  the  first  term  of  the  numera- 
tor, and  the  denominators  are  positive  ;  hence  both  ml  and  m2  are 
real  and  negative. 

Making  use  of  (g)  and  (^),  we  can  now  obtain  from  (</)  and  (e) 
the  values  of  AjB  satisfying  (a)  and  (£).  Thus,  putting  m  =  ml 
in  (d)  and  (e\  and  denoting  the  resulting  value  of  AjB  by  Alj  Blt 
we  obtain 


(i) 
and  similarly,  putting  m  =  m2, 


Thus  the  general  solutions  of  (in),  (112),  (#),  and  (^)  are 

(k) 


and 

72  =  BJ**  +  ^/Wi2< 


where  ^j  and  A2  are  to  be  determined  from  the  initial  conditions. 
A.  The  circuit  containing  M/*  is  suddenly  closed.     The  initial 
conditions  are  7t  =  72  =  o  when  t  =  o,     Hence,  from  (z),  (/), 
(£),  and  (/),  we  obtain 


and 


7,  =  - 


i      \ 

*          I        / 


+  i  \  emv 

(H3) 


The  relations  between  7j  and  f2  and  the  time  after  closing  the 
circuit   i  are  shown  approximately  for  the  general  case  in  Fig. 


396 


ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 


123.     7j  approaches  asymptotically  a  final  value  "^/Rv  and  J2 
has  a  maximum  at  the  time  given  by  dl^dt  =  o. 

The  total  charge  traversing  circuit  2,  obtained  either  by  inte- 
grating (114)  from  the  time  t  =  o  to  the  time  t  =  infinity,  or 
from  (12)  directly,  is 


q  = 

=  -  MV/R&  =  -  MI/R2 
if  /denotes  the  final  current  ^  JRY  in  circuit  I. 


Fig.  123. 


B.  The  electromotive  force  ^  is  suddenly  cut  out  of  circuit  I 
(without  opening  the  circuit).  The  values  of  the  currents,  in 
terms  of  II  and  72,  given  in  (113)  and  (114),  are 


and  ',"--',  (U7) 

/  now  denoting  the  time  reckoned  from  the  instant  of  cutting  out 
the  electromotive  force. 

50.  Case  II.    The  Electromotive  Force  is  a  Simple  Harmonic 
Function  of  the  Time,  "^0  cos  pt.     In  this  case  we  have 

LjlIJdt  +  Mdl^dt  +  RJi  =  ¥0  cos  //  (  1  1  8) 

and  Ljtljdt  +  Mdljdt  +R2?2=o  (119) 


ELECTROMAGNETIC    INDUCTION.  397 

The  general  solution  of  these  equations  is  that  given  in  (k) 
(for  //)  and  (/)  added  to  a  particular  solution  now  to  be  obtained. 

Since  the  applied  e.m.f.  is  harmonic,  we  may  assume  that  the 
currents  to  which  it  gives  rise  will  also  be  harmonic  in  the  same 
period.  Hence  we  put 

/!=  A  cos  (//  —  0J  (120) 

and 

72  =  B  cos  (pt  -  02)  (121) 

where  A,  B,  6l  and  02  are  constants  to  be  determined.  To  deter- 
mine these  constants,  substitute  (120)  and  (121)  in  (118)  and 
(119),  and  equate  to  zero  the  coefficients  of  cos  pt  and  sin  // 
separately,  since  the  resulting  equations  are  true  for  all  values 
of  t.  Thus,  putting 


L'  =  L,  -  pWL,l(Rf  +  /Z/)  (122) 

and 

R'  =  R,  +  /M*X2/(X*  +/L?)  (123) 


we  find,  as  conditions  that  (120)  and  (121)  may  be  solutions  of 
(1  1  8)  and  (i  19), 

l  (124) 


•       (125) 

=  tan-1  /£'/•#'  (126) 

(127) 


which,  inserted  in  (120)  and  (121),  give  the  particular  solution 
sought. 

The  terms  given  by  (k)  and  (/)  become  negligible  in  a  very  short 
time,  hence  after  this  time  (120)  and  (121)  are  the  complete  solu- 
tion of  (118)  and  (119). 

On  comparing  (120)  and  (124)  with  (103),  we  see  that  the 
current  in  circuit  I  is  the  same  as  it  would  be  if  circuit  2  were 
not  present  and  the  resistance  of  circuit  I  were  increased  from 
R,  to  R'  ,  and  its  inductance  diminished  from  L^  to  L'. 


39$  ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

When  /2Z2  is  great  in  comparison  with  R2,  (122)  and  (123) 
become  approximately 

L'  =  A  -  M*/L2     and  R'  =  ^  +  X2M2/L* 

If  in  addition  the  coils  are  wound  together  so  that  the  mag- 
netic leakage  is  zero,  or  L^L^  =  M2,  approximately,  L1  becomes 
approximately  zero.  In  this  case,  Ol  =  o,  the  current  in  circuit 
I  being  in  phase  with  the  e.m.f.,  and  02  =  TT,  so  that  the  current 
in  circuit  2  lags  behind  the  current  (and  e.m.f.)  in  circuit  I  by 
half  a  period. 

If  either  circuit  contains  in  series  a  condenser  of  capacity  S, 
the  solution  is  obtained,  as  follows  from  §  44,  by  writing  in  place 
of  L^  or  Z2,  Z:  —  i/S/2  or  L2  —  i/S/2,  in  the  expressions  for  A, 
B,  0V  and  0y 

If  the  magnetic  field  contains  iron,  or  other  substance  in  which 
hysteresis  occurs,  a  simple  harmonic  applied  e.m.f.  will  not  de- 
velop a  simple  harmonic  current.  For  the  current  is  propor- 
tional to  the  magnetic  intensity  Hy  by  the  first  law  of  circuita- 
tion,  and  the  induced  part  of  the  total  e.m.f.  is  proportional 
to  the  rate  of  change  dB  jdt  of  the  magnetic  induction  B.  But 
if  dBjdt  is  a  simple  harmonic  function  of  the  time,  then,  if  there 
is  hysteresis,  //"cannot  be,  as  is  evident  from  Fig.  114.  This  of 
course  applies  to  single  circuits  as  well. 

51.  The  Ratio  of  Transformation  of  Two  Coils  for  Which  L^L2  = 
M2  and  Rl  =  R2  =  o,  approximately.  Let  two  coils  I  and  2  be 
so  wound  that  all  the  magnetic  flux  <3>  threads  every  turn  of  both 
coils  (magnetic  leakage  =  o).  Let  coil  I  contain  nv  turns  and 
coil  2  n2  turns  ;  then  the  flux  through  coil  I  will  be  Nv  =  nfl* 
and  the  flux  through  coil  2  will  be  N2  =  nfi. 

The  e.m.f.  applied  between  the  terminals  of  coil  I  will  be 

^  =  R^  +  dNjdt 
and  the  e.m.f.  between  the  terminals  of  coil  2  will  be 


ELECTROMAGNETIC    INDUCTION.  399 

the  same  direction  being  chosen  as  positive  around  both  circuits. 
Hence  if  R^  and  R.J2  are  negligible  in  comparison  with  the 
electromotive  forces  of  induction,  we  have,  in  magnitude, 

^/¥2  =  nld<$>ldtln2d$>ldt  =  njn2  (128) 

which  is  called  the  ratio  of  transformation  of  e.m.f.s  of  the  two 
coils. 

Rl  and  R2  above  denote  only  the  resistance  of  the  coils,  or 
those  parts  of  circuits  I  and  2  threaded  by  the  tube?  of  magnetic 
induction  common  to  both  circuits. 

The  power  supplied  to  the  system  by  the  e.m.f.  ^  is  ^/j,  and 
the  power  supplied  to  that  part  of  circuit  2  outside  the  coil  is 
^2/2.  When  no  energy  is  dissipated  in  the  coils  we  have 

\Er  /  _  \Er  / 

^Iyi~     *2J2 

whence  7^7,  =  ^/^  =  njn^  (129) 

which  is  the  ratio  of  transformation  of  currents  for  the  case  con- 
sidered. 

52.  Electromagnetic  Repulsion  Between  Two  Circuits.  When 
an  alternating  current  is  induced  in  a  circuit  (2)  owing  to  the 
circulation  of  an  alternating  current  in  a  neighboring  circuit  (i) 
connected  with  an  alternating  current  generator,  a  force  is 
developed  between  the  two  circuits,  positive  or  repulsive  when 
the  two  currents  have  opposite  directions,  and  negative  or 
attractive  when  the  two  currents  have  the  same  direction.  We 
proceed  to  show  that  the  average  force  is  one  of  repulsion. 

For  the  sake  of  simplicity  let  the  two  circuits  be  circular  with 
their  afces  coincident,  and  let  circuit  (2)  consist  of  a  single  turn 
of  wire.  Let  the  magnetic  flux  through  circuit  (2)  be  a  har- 
monic function  of  the  time, 

N=A  cos  nt 

Then  the  outward  radial  component  of  the  magnetic  induction  at 
all  points  of  circuit  (2)  will  be 


400          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

Br  =  a  cos  nt 
where  a  is  a  constant 

The  e.m.f.  induced  in  circuit  (2)  will  be 

^  =  _  dN]dt  =  nA  sin  nt 
and  the  current  in  circuit  (2), 

/=  C  sin  (nt—  0) 

where  C  and  6  are  positive  constants  (0  <  90°). 

The  force  upon  circuit  (2)  in  the  direction  I  2  is  therefore 

F=  —  VIBr  x  length  of  circuit  (2) 
==  —  cos  ;//  •  sin  (nt  —  6)  x  positive  constant 

When  cos  nt  and  sin  (nt  —  0)  have  the  same  sign,  or  opposite 
signs,  the  force  upon  coil  (2)  is  thus  an  attraction  toward  (i), 
or  a  repulsion  from  (i),  respectively.  On  platting  cos  nt, 
sin(nt  —  0),  and  their  product,  which  is  proportional  to  —  F,  as 
functions  of  the  time,  it  will  be  seen  that,  owing  to  the  lag  of  the 
current,  the  average  force  is  a  repulsion  between  the  circuits. 
The  same  result  follows  from  integrating  Fdt  throughout  a 
complete  period,  the  average  force  being 

i    CT  i    CT 

—  I    Fdt  =s  —  —  I    cos  nt  sin  (nt  —  0)  •  dt  x  positive  constant 

*  «/o  *•  J o 

Analogous  theory  applies  to  a  great  variety  of  cases,  one  of 
the  simplest  being  that  of  a  light  coil,  or  disc,  of  copper,  alu- 
minium, or  other  good  conductor  placed  horizontally  over  an 
electromagnet  with  coils  horizontal.  When  the  magnet  is  power- 
fully excited  by  an  alternating  current  the  disc  or  coil  is  thrown 
violently  up  into  the  air.  Another  interesting  case  is  that  of  a 
horizontal  copper  disc  mounted  on  a  pivot  and  placed  eccentric- 
ally over  the  electromagnet,  with  a  second  disc,  held  likewise 
horizontal,  near  by.  Owing  to  the  asymmetry  thus  introduced, 
the  pivoted  disc  rotates  continuously  while  the  magnet  is  power- 
fully excited. 

53.  The  Comparison  of  a  Mutual  Inductance  and  a  Resistance 
One  (i)  of  the  two  coils,  of  mutual  inductance  M,  is  connected 


ELECTROMAGNETIC    INDUCTION.  40 1 

in  circuit  with  a  constant  battery  B  through  a  key  Ky  while  the 
other  (2)  is  connected  in  circuit  with  a  ballistic  galvanometer,  the 
total  resistance  of  circuit  (2)  being  R. 

When  a  current  /is  suddenly  started  or  stopped  in  coil  (i) 
by  closing  or  opening  the  key  Ky  a  charge  q  =  MIj R  circulates 
through  coil  (2)  and  produces  an  angular  throw  0  of  the  needle 
such  that  MIjR  =  H  jy  £TT  •  sin  \Q 

or  M/R=H/GS-T/7r-sml0  (130) 

Circuit  (2)  is  now  cut  at  some  point  and  the  resulting  ter- 
minals connected  to  two  points  on  circuit  (i)  with  a  resistance  r, 
very  small  in  comparison  with  the  total  resistance  of  circuit  (i), 
between  them.  This  will  not  sensibly  affect  the  current  in  cir- 
cuit (i),  but  will  cause  a  steady  current 

r/(r +*)•/  =  H/G-f[0')(See  §  33,  XII.)         (131) 

to  traverse  the  galvanometer,  producing  the  steady  deflection  Of ' . 
Eliminating  H/  GI  from  (130)  by  (131),  we  obtain,  finally, 

M\R  =  r/(r  +  R)  •  T/TT  •  sin  £0/^(0')  (132) 

6,  F(6'),  T,  and  the  ratio  of  r  to  r  +  R  being  observed,  M\R  is 
given  in  absolute  measure  by  (132). 

Absolute  Determination  of  a  Resistance.  From  the  dimensions 
of  the  coils,  if  circular,  toroidal,  or  rectangular,  M  can  be  calcu- 
lated ;  hence  the  method  affords  an  absolute  determination  of  a 
resistance.  See  Glazebrook,  Phil.  Trans.,  1883,  for  an  important 
investigation  in  which  this  method  was  adopted. 

The  Comparison  of  Two  Mutual  Inductances.  By  sending  the 
same  current  in  succession  through  the  primaries  of  two  coils 
and  connecting  the  secondaries  in  succession  through  the  same 
galvanometer,  the  total  resistances  of  the  secondary  circuits  in 
the  two  cases  being  Rl  and  R2,  the  two  mutual  inductances  Ml 
and  Mv  and  the  two  deflections  6l  and  02,  the  ratio  of  the  two 
mutual  inductances  can  be  obtained  from  the  relation 

MJMt  =  R,  sin  Jtf,/*,  sin  102  (133) 


402  ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

By  putting  an  adjustable  resistance  in  one  circuit,  or  in  each 
circuit,  and  changing  the  resistance  until  6l  =  @2,  we  have 

MJM^RJR,  (134) 

By  connecting  the  two  secondaries  in  series  with  the  galvano- 
meter permanently,  and  passing  the  same  current  in  succession 
through  the  two  primaries,  we  have 

MJM^sml-OJsm^  (135) 

The  Comparison  of  Resistances,  The  method  can  also  be  em- 
ployed for  the  comparison  of  two  resistances  in  the  secondary 
circuits.  In  this  case 

^/^sin^/sini^  (136) 

54.  The  Measurement  of  Magnetic  Induction  by  the  Ballistic 
Method.  A  small  coil  consisting  of  n  turns  of  fine  insulated  wire 
uniformly  wound  in  parallel  planes,  the  mean  area  enclosed  by 
a  single  turn  being  5,  is  connected  by  twisted  wires  to  the  ter- 
minals of  a  ballistic  galvanometer.  The  coil  is  introduced  into 
the  magnetic  field  at  the  place  where  the  induction  is  to  be  de- 
termined with  the  planes  of  its  turns  perpendicular  to  the  induc- 
tion, and  the  galvanometer  needle  is  brought  to  rest.  Then  the 
coil  is  suddenly  jerked  out  of  the  magnetic  field,  and  the  result- 
ing deflection  6  of  the  galvanometer  needle  is  read.  Let  B 
denote  the  mean  induction  perpendicular  to  the  planes  of  the 
coil's  turns  in  the  part  of  the  field  in  which  the  coil  was  placed, 
and  let  R  denote  the  resistance  of  the  circuit.  Then,  by  (12), 
and  (49)  XII.,  the  charge  traversing  the  circuit  and  producing 
the  deflection  0  is 

q  =  nSBfR  =  H  T/TrG  •  sin  -|  0 
from  which 

B  =  RttT/irnSG-  sin  J0  (137) 

If  fji  (sensibly  equal  to  unity  in  nearly  all  cases)  is  known,  the 
magnetic  intensity  can  be  found  by  dividing  B  by  p. 


ELECTROMAGNETIC    INDUCTION. 


403 


'55.  Maxwell's  Method  of  Comparing  Two  Mutual  Inductances. 

In  the  practise  of  this  method  (Fig.  124)  two  of  the  coils  (the 
primaries),  one  from  each  pair,  are  connected  in  series  with  the 
battery  through  a  key  K,  and  the  terminals  of  each  of  the  other 
coils  (the  secondaries)  are  connected  to  the  terminals  of  a 
galvanometer  G  through  an  adjustable  resistance,  connections 
being  so  arranged  that  the  discharges  of  the  two  secondaries,  on 


Fig.   124. 

opening  or  on  closing  K,  when  tested  separately,  traverse  the 
galvanometer  in  opposite  directions.  Then  the  resistances  in  the 
secondaries  are  adjusted  until  the  galvanometer  needle  remains 
undeflected,  i.  e.t  until  the  total  discharge  through  the  galvanome- 
ter reckoned  in  one  direction  is  zero,  when  K  is  opened  or  closed. 
Then,  if  R^  and  R2  denote  the  adjusted  resistances  from  A  to  B 
through  C  and  from  A  to  B  through  D,  respectively, 

MJM^RJR%  (I38) 

To  prove  this,  let  7  denote  the  current  through  the  battery  and 
primaries,  Iv  72,  and  73  =  7X  —  72  the  currents  from  A  to  B  through 
C,  from  A  to  B  through  D,  and  from  A  to  B  through  the  gal- 
vanometer, at  the  time/;  and  let  Lv  L2,  and  Z3  denote  the  in- 
ductances of  ACB,  ADB,  and  AGB,  and  J?3  the  resistance  of 
AGB.  Then  the  impressed  e.m.f.  from  A  to  B  at  the  time  /  is 

M,dl\dl  -  L.dljdt  -  RJ,  =  V(/,  -  Wdt  +  R,(T,  -  72) 


404          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 
and  the  impressed  e.m.f.  from  B  to  A  at  the  time  /  is 


Multiplying  each  equation  by  dt  and  integrating  from  the  time 
of  closing  the  circuit  (t  =  o)  to  the  time  at  which  the  battery 
current  becomes  steady  (t  =  ^),  remembering  that  the  initial 

c* 

and  final  values  of  7T  and  /2  are  zero  and  that  I     (7X  —  I^dt,  the 

total  discharge  through  the  galvanometer,  is  zero,  and  denoting 
the  steady  value  of  the  battery  current  by  /0,  we  have 


-  R2  C  I2dt  =  o 

*/0 

from  which  (138)  immediately  follows,  since 

r/^=  ["//' 

t/O  t/0 

The  Comparison  of  Resistances.  We  can  vary  the  method 
slightly  for  the  comparison  of  two  resistances.  Suppose  the 
balance  given  by  (138)  effected.  Introduce  an  unknown  re- 
sistance X  into  the  branch  ADB,  and  balance  by  adding  a 
known  resistance  R  to  R  Then 


Combining  this  equation  with  (138)  we  find 

(139) 


If  X  and  RJ  are  both  known,  (139)  gives  MJM1  without  a 
knowledge  of 


56.  Maxwell's  Method  of  Comparing  an  Inductance  and  a  Capac- 
ity. The  given  coil,  with  inductance  L  and  resistance  R,  is  con- 
nected up  in  a  Wheatstone's  bridge  with  three  non-inductive 


ELECTROMAGNETIC    INDUCTION. 


405 


resistances  Pt  Q,  T,  Fig.  125,  and  a  resistance  balance  for  steady 
currents  is  obtained  in  the  usual  way  by  adjusting  the  resistances 
until  the  galvanometer  needle  remains  undeflected  when  K2  is 


Fig.  125. 

closed  after  Kr     Then,  if  Vab  denotes  the  fall  of  potential  from 
A  to  B,  etc.,  and  /the  current  through  ABC, 

.        VJ  V*  =  RffTf-  RIT=  VJ  V,c  =  P/Q 

During  the  variable  state  of  the  currents  just  after  closing  or 
opening  Kv  however,  K2  being  open, 


Hence,  if  K^  is  opened  or  closed  while  Kz  is  closed,  the  gal- 
vanometer needle  will  be  deflected. 

If,  however,  a  condenser  is  introduced  as  a  shunt  to  DC,  as 
shown  by  the  dotted  lines,  a  part  of  the  current  through  P  will 
be  shunted  into  this  condenser  during  the  rise  of  the  current  after 
the  closing  of  Kv  thus  reducing  the  current  through  Q  and 
increasing  the  ratio  V^j  Vdc  ;  and  during  the  decrease  of  the  cur- 
rent after  the  opening  of  K^  the  discharge  of  the  condenser  will 
increase  the  current  through  Q  and  decrease  the  current  through 
Pt  thus  diminishing  the  ratio  Vadj  Vdc.  Hence,  since  the  law  of 
the  increase  or  decrease  of  the  induced  current  in  an  inductive 
resistance  with  the  time,  and  the  law  of  the  increase  or  decrease 


406         ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

of  the  charging  current  of  a  condenser  with  the  time,  are  identi- 
cal, §§  41-42,  an  exact  balance  for  both  transient  and  steady 
currents  can  be  obtained  by  using,  with  given  inductance  and 
resistances,  a  condenser  of  a  particular  capacity.  By  readjusting 
the  non-inductive  resistances,  however,  a  balance  for  both  steady 
and  transient  currents  may  be  obtained  for  any  inductance  and 
capacity.  Thus,  if  R  and  P  are  fixed,  the  ratio  of  the  effect  of 
the  inductance  to  that  of  the  capacity  is  decreased  by  increasing 
Q  and  T  and  keeping  always  Q/T=  PjR.  When  this  double 
balance  has  been  attained,  then,  as  shown  below, 

L/S=RQ  =  PT  (140) 

Anderson's  Modification.  To  avoid  the  necessity  of  this  tedious 
process  of  readjustment  and  trial,  an  additional  non-inductive 
resistance  W  may  be  inserted  between  D  and  the  condenser  and 
galvanometer,  Fig.  126.  This  will  not  affect  the  balance  for 


Fig.  126. 

steady  currents,  but  will  enable  the  effect  of  the  condenser  on 
the  balance  for  variable  currents  to  be  altered.  After  the  balance 
for  steady  currents  has  been  attained,  the  resistance  IV  is  altered 
until  the  balance  is  good  for  transient  currents  also.  Then 

L/S=  W(R+  T)+PT  (141) 

W  thus  increases  the  effect  of  the  condenser.  If  the  condenser 
has  too  great  an  effect  when  W '  =  o,  PT(or  QR)  must  be  de- 
creased in  the  balance  for  steady  currents. 

To  establish  (141),  let  xy  x,y  -f  z,y,  and  z  denote  the  currents 
in  the  branches  R,  T,  P,  Q,  and  W,  respectively,  at  the  time  /  after 
closing  or  opening  Kv  the  double  balance  having  been  attained. 


ELECTROMAGNETIC    INDUCTION.  407 

Then,  since  the  fall  of  potential  along  the  path  ABE  is  equal 
to  that  along  the  path  ADE,  we  have 

Rx  +  Ldxjdt  =  P(y  +  z)  +  Wz  (a) 

In  the  same  way  the  fall  of  potential  from  D  to  C  through  Q 
is  equal  to  that  from  D  to  C  through  W  and  the  condenser  ; 
that  is 

Qy=  W*  +    q,+dtlS  (8) 


where  q^  denotes  the  initial  charge  of  the  condenser. 

Likewise,  the  fall  of  potential  from  E  to  C  through  the  con- 
denser is  equal  to  that  from  E  to  C  through  the  galvanometer 
and  7";  or 

=TX  (t) 


Eliminating  x  and  y  from  (a)  by  (b)  and  (c),  differentiating 
with  respect  to  t,  and  equating  to  zero  separately  the  coefficients 
of  z  and  dzjdt  (since  the  equation  holds  for  all  values  of  z  and 
dzjdt,  including  zero),  we  obtain  the  conditions  of  a  double 
balance  : 

RjT=PQ  (d) 

the  condition  for  a  balance  for  steady  currents  ;  and  (141),  viz., 

L/S=(R  +  T)W+PT 
which  reduces  to  (140)  when  W  =  o. 

Russell's  Modification.  If  a  given  inductance  is  to  be  com- 
pared with  an  adjustable  standard  capacity,  or  if  a  capacity  is  to 
be  compared  with  a  standard  inductance  which  can  be  adjusted 
to  different  values  without  an  alteration  of  resistance,  the  balance 
for  steady  currents  is  first  effected,  then  the  galvanometer  circuit 
is  closed  and  the  standard  capacity  or  inductance  altered  until 
the  galvanometer  needle  remains  undeflected  when  the  battery 
key  is  opened  or  closed. 


408 


ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 


57.  Maxwell's  Method  of  Comparing  Inductances,  The  two 
coils  whose  inductances  7X  and  Z2  are  to  be  compared  are  joined 
up  with  four  non-inductive  adjustable  resistances  and  a  battery 
and  galvanometer  as  shown  in  Fig.  127.  (The  connections  of 


Fig.  127. 

the  battery  and  galvanometer  may  be  interchanged.)  Rv  R^ 
etc.,  denote  the  total  resistances  in  the  branches  AB,  BC,  etc. 
The  resistances  are  varied  until  the  galvanometer  shows  no  deflec- 
tion for  steady  currents  (K2  closed  after  K^)  or  for  transient  or 
variable  currents  (Kv  closed  after  K2  or  opened  before  K^.  Then 

L,IL,  =  R,IR,  =  R,IR,  (,42) 

For,  when  such  a  double  balance  has  been  effected,  the  voltage 
from  A  to  B  is  equal  to  the  voltage  from  A  to  D  at  any  time  /  ; 
that  is, 


where  /x  and  73  denote  the  currents  through  ABC  and  ADC  re- 
spectively. In  like  manner  the  voltage  from  B  to  C  is  equal  to 
the  voltage  from  D  to  C,  that  is 


Combining  these  two  equations  and  dividing  the  resulting  equa- 
tion by  73,  we  obtain 


L^dl^dt  = 


j  + 


ELECTROMAGNETIC    INDUCTION. 


409 


Since  this  equation  is  true  for  all  values  of  the  current  II  and 
its  rate  of  change,  including  zero,  we  have 


which  is  the  condition  for  a  balance  with  steady  currents,  and 


which  is  the  additional  condition  for  a  balance  for  variable  cur- 
rents.     From  these  equations  (142)  follows  immediately. 

Standard  coils  are  constructed  whose  inductances  can  be  varied 
within  wide  limits  without  an  alteration  of  resistance.  If  one  of 
the  coils  to  be  compared  is  a  standard  of  this  kind,  a  balance  for 
steady  currents  is  first  obtained  in  the  ordinary  way ;  then,  with- 
out altering  the  resistance  in  any  part  of  the  network,  a  balance 
for  transient  currents  is  made  by  altering  the  inductance  of  the 
standard.  See  §  18. 

58.  Gary  Foster's  Methods  of  Comparing  a  Mutual  Inductance 
and  a  Capacity.  Let  connections  be  made  as  in  Fig.  128,  vS  and 


Fig.  128. 

M  denoting  the  capacity  and  mutual  inductance,  R^  the  total  re- 
sistance from  A  through  K^  to  B,  and  R2  the  resistance  between 
the  points  A  and  C,  both  resistances  being,  at  least  in  part,  ad- 
justable. If  K^  is  open  and  K2  closed,  then  when  K  is  opened 
or  closed  a  charge  SR2I  will  traverse  the  galvanometer,  /  de- 
noting the  steady  value  of  the  current  in  the  battery  circuit.  If 
K2  is  open  and  Kv  closed,  a  charge  MIj(Rl  +  g)y  where  g  de- 
notes the  galvanometer  resistance,  will  traverse  the  galvanometer. 


410          ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

By  reading  the  two  galvanometer  deflections  the  ratio  of  M  to  6* 
may  therefore  be  obtained. 

In  Gary  Foster's  null  method  connections  are  so  made  that 
on  closing  K  or  on  opening  K  the  capacity  and  inductance  dis- 
charges are  in  opposite  directions  through  the  galvanometer. 
Then  Kv  and  Kz  are  both  closed,  and  the  resistances  Rl  and  J?2 
are  adjusted  until  on  opening  or  closing  K  there  is  no  deflection 
of  the  galvanometer  needle,  showing  that  the  total  discharge 
through  .the  galvanometer  is  zero.  Then 

RIS 


g  not  entering  the  expression.      Hence 

MjS^R.R,  (143) 

The  demonstration  is  left  to  the  reader,  who  should  refer  to 

§55. 

59.  Brillouin's  Modification  of  Maxwell's  Method  of  Comparing- 
the  Mutual  Inductance,  M,  of  Two  Coils  with  the  Self-Inductance, 
L,  of  One  of  Them.  The  coil  with  inductance  L  and  resistance  R 


Fig.  129. 


is  connected  up  with  three  non-inductive  resistances  P,  Q,  T  in 
a  Wheatstone's  bridge,  Fig.  129,  and  the  other  coil  is  connected 
through  an  adjustable  resistance  to  the  points  A  and  C.  Balance 
is  first  obtained  for  steady  currents  in  the  usual  way,  then  the 


ELECTROMAGNETIC    INDUCTION.  4^ 

resistance  5  of  the  branch  A SC  is  adjusted  until  the  galvanometer 
shows  no  deflection  when  the  galvanometer  circuit  is  closed  and 
the  battery  key  is  opened  or  closed.  In  order  that  the  mutual 
inductance  of  the  two  coils  may  thus  neutralise  the  effect  of  the 
self-inductance  of  one  of  them,  the  two  coils  must  be  so  con- 
nected that  their  currents  flow  always  in  opposite  directions 
around  the  tubes  of  induction  which  thread  them.  Then 

L\M~  (R  +  T)/S  (144) 

For  the  increase  of  voltage  from  A  to  B  due  to  the  self-induc- 
tance is  L  dxjdt,  and  the  decrease  in  the  voltage  due  to  the 
mutual  inductance  is  Mdzjift,  where  x  and  z  denote  the  currents 
through  ABC  and  ASC  respectively.  Hence,  when  the  balance 
for  transient  currents  is  attained, 

Ldxldt=Mdz\dt 

Integrating  through  the  time  t  in  which  the  currents  rise  from 
zero  to  their  steady  values  XQ  and  #0>  or  in  which  they  drop  from 
their  steady  values  to  zero,  we  have 

L  I   dxjdt  dt  =  LxQ  =  M  I  dzjdtdt  = 

t/O  «/0 

Hence  L/M=  *0/*0  =  (R  +  T)/S 

60.  The  Comparison  of  a  Capacity  with  a  Resistance.  A  ca- 
pacity may  be  compared  with  a  resistance  by  a  method  exactly 
analogous  to  that  of  §  53,  as  shown  in  §  36,  XII.  The  following 
null  method,  due  to  Maxwell,  is,  however,  much  to  be  preferred. 

One  branch  of  the  Wheatstone's  bridge,  as  the  branch  23,  Fig. 
76  is  cut  and  a  condenser  AB  inserted,  the  plate  B  being  con- 
nected by  a  wire  of  negligible  resistance  to  the  point  3,  and  the 
plate  A  being  connected  in  the  same  way  with  a  rapidly  and 
uniformly  moving  commutator,  which  puts  it  alternately  into 
electrical  contact  with  the  point  3  and  the  point  2.  When  A  is 
connected  to  3  the  condenser  is  short-circuited  and  the  galvanom- 


412          ELEMENTS    OF   ELECTROMAGNETIC    THEORY. 

eter  is  traversed  by  a  current  in  the  direction  24  ;  while  during 
a  part  of  the  time  the  plate  A  is  connected  with  2  the  galvanom- 
eter is  traversed  by  a  current  in  the  opposite  direction  42,  the 
plate  A  being  charged  through  the  galvanometer  and  the  branch 
1  2.  If  the  period  of  the  galvanometer  needle  (or  coil)  is  great  in 
comparison  with  the  period  of  the  commutator,  a  steady  deflec- 
tion will,  in  general,  result.  By  suitably  adjusting  the  resistances 
a,  b,  and  d,  however,  the  average  current  through  the  galvanom- 
eter, reckoned  as  positive  in  one  direction  and  negative  in  the 
other,  may  be  made  zero,  when  the  galvanometer  will  show  no 
deflection  whether  the  battery  is  connected  to  the  bridge  or 
not.  This  is  the  only  adjustment  to  be  made  in  the  practise  of 
the  method. 

When  this  balance    has    been    effected,  the  average  voltage 
from  I  to  2  is  equal  to  the  average  voltage  from  I  to  4,  i.  e.,  to 


if  ^  denotes  the  e.m.f.  of  the  battery,  and  if  the  battery  resistance 
is  negligible  (a  condition  easy  to  attain)  in  comparison  with  that 
of  the  bridge  (from  point  I  to  point  3).  Hence  the  average  value 
of  the  current  in  the  branch  1  2  is 


and  this  is  equal  to  the  average  charging  current  of  the  con- 
denser, since  the  average  current  through  the  galvanometer  is 
zero. 

The  voltage  of  the  condenser  when  fully  charged  is  the  volt- 
age between  the  points  I  and  2  when  the  current  in  every  part  of 
the  bridge  has  reached  its  steady  state.  This  voltage  is  readily 
seen  to  be 


d(a  +b  +  g) 

Hence,  if  n  denotes  the  frequency  of  the  commutator,  or  the 
number  of  times  the  condenser  is  charged  per  second,  and  5  the 


ELECTROMAGNETIC    INDUCTION.  413 

capacity  of  the  condenser,  the  average  value  of  the  charging  cur- 
rent of  the  condenser  is 

bg  +  d(a  -f  b  4-  g) 
b(a  +  g)  +  d(a+  b  +  g) 

Equating  the  two   expressions  for  /  and  solving  for  S,  we 
obtain 


~  na(b  - 


When  the  condenser  has  a  guard  ring,  this  method  cannot  be 
used  without  an  inconvenient  modification,  f  A  much  simpler 
method,  however,  equal  in  accuracy,  can  be  used  in  all  cases.  A 
galvanometer  with  two  independent  coils  acting  on  the  same 
needle  (differential  galvanometer)  has  the  intermittent  condenser 
current  sent  through  one  coil  and  a  steady  current  from  the 
battery  in  the  opposite  direction  through  the  other.  The  deflec- 
tion is  reduced  to  zero  by  suitably  adjusting  the  resistance  con- 
nected with  this  second  coil.  This  method,  due  to  Klemencic, 
has  been  used  with  great  precision  by  Himstedt  %  and  by  H. 
Abraham  §  in  the  determination  of  a  (XIV.,  §  4). 

Comparison  of  Capacities.  These  methods  also  serve  admira- 
bly for  the  comparison  of  capacities,  only  the  ratios  of  the  re- 
sistances being  necessarily  known. 

61.  The  Comparison  of  an  Inductance  with  a  Resistance,  The 
coil  AB,  with  inductance  L,  is  connected  up  with  three  non-in- 
ductive resistances  P,  Q,  and  T,  a  galvanometer,  and  a  constant 
battery,  as  shown  in  Fig.  125  (the  dotted  lines  being  annulled), 
and  a  balance  for  steady  currents  is  effected  in  the  usual  way. 
With  the  key  K2  closed,  Kl  is  then  either  opened  or  closed,  when 
the  flux  through  the  coil  will  change  from  LI  to  o  or  from  o  to 

*H.  Abraham,  Ann.  de  Chim.  et  de  Phys.,  Vol.  27,  1892. 

|  For  an  important  investigation  in  which  the  method  was  modified  and  used  with 
a  guard  ring  condenser,  see  Thomson  and  Searle,  Phil.  Trans.,  A,  1890. 
\Wied.  Ann.,  Vol.  35,  1888. 
3  Loc.  cit. 


414          ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

LI,  I  denoting  the  steady  value  of  the  current  in  AB.  Owing 
to  the  change  of  flux  and  the  e.m.f.  developed  thereby  in  the 
coil  AB,  a  charge  q  proportional  to  Z/will  traverse  the  galva- 
nometer, producing  an  angular  throw  0,  such  that 


IT 


where  K  is  a  constant  depending  on  the  resistances. 

The  balance  for  steady  currents  is  then  disturbed  by  increasing 
or  decreasing  the  resistance  R  of  the  branch  AB  by  a  very  small 
quantity  A^.  If  /'  denotes  the  new  value  of  the  steady  current 
in  AB,  an  e.m.f.  If&R  will  thus  exist  in  the  branch  during  the 
steady  state,  and  a  steady  deflection  Q'  of  the  galvanometer 
needle  will  result,  such  that 


K  being  the  same  constant  occurring  in  the  previous  equation. 
Eliminating  K  from  the  two  equations  and  solving  for  L,  we 

have 

/'     Tsin  10 


The  ratio  /'//  of  the  final  and  initial  values  of  the  steady 
current  in  AB  can  be  calculated  from  the  resistances  in  the 
bridge.  It  is  obvious  that  ordinarily  the  ratio  will  be  sensibly 
equal  to 


which  is  very  nearly  unity. 

The  method  was  originated  by  Maxwell,  and  was  first  used 
with  precision  in  an  important  investigation  by  Lord  Rayleigh 
(Phil.  Trans.,  Part  II.,  1882). 


CHAPTER    XIV. 
UNITS   AND    DIMENSIONS. 

1.  The  Electrostatic  Systems  of  Units,  (i)  The  rational  electro- 
static system.  The  rational  electrostatic  unit  charge,  fully  dis- 
cussed in  Chapter  L,  is  defined  by  the  equation 

q~(#ircDFy  (I) 

where  L  and  .Fare  expressed   in  c.g.s.  units  and  c  is  expressed 
in  terms  of  the  permittivity  of  free  aether  (r0)  as  unit  permittivity. 

From  this  fundamental  definition  and  the  further  definition  that 
all  the  equations  hitherto  developed  (except  those  specified  as 
belonging  to  irrational  systems)  hold  good  for  all  rational  systems 
of  units  (electromagnetic  as  well  as  electrostatic),  the  definitions 
of  the  rational  electrostatic  units  of  all  the  other  electrical  quan- 
tities follow.  Thus  the  definition  of  the  RES  unit  current  is 
given  by  the  equation  /=  qjt,  /being  expressed  in  RES  unit 
current  when  q  is  expressed  in  the  RES  unit  charge  and  t  in  the 
c.g.s.  unit  time  ;  similarly,  the  definition  of  the  RES  unit  mag- 
netic intensity  then  follows  from  the  relation  H  =  m.m.f.  =  H 
X  27rd=I(§  14  or  15,  XII.);  the  definition  of  the  RES  unit 
magnetic  pole  strength  then  follows  from  the  equation  m  =  FjH\ 
and  then  the  definition  of  the  RES  unit  magnetic  inductivity  from 
the  equation  /x  =  nfi j AfrrD'F ;  etc. 

(2)  The  common  or  irrational  electrostatic  system.  If,  however, 
while  the  units  of  permittivity,  force,  and  length  remain  un- 
changed, A,  equation  (ir),  L,  is  put  equal  to  unity  instead  of  477-, 
we  obtain,  as  the  definition  of  the  (irrational  or  common)  electro- 
static unit  charge,  the  equation 

q=(cL^  (2) 

415 


416         ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

On  comparing  this  equation  with  (i)  we  see  that  the  electro- 
static unit  just  denned  is  equal  to  (477-)*  times  the  rational  unit. 

The  system  of  ejectrical  units  built  up  from  this  unit  charge 
as  fundamental  unit  in  a  manner  exactly  similar  to  that  in  which 
the  rational  system  is  built  up  from  the  RES  unit  charge,  with 
the  exception  of  a  few  units,  such  as  intensity  of  electrisation  or 
magnetisation,  which,  as  stated  in  appropriate  places  in  the  text, 
are  differently  denned,  is  called  the  electrostatic  (ES)  system  of 
units. 

2,  The  Electromagnetic  Systems  of  TJnits.  (i)  The  rational 
electromagnetic  system.  The  rational  electromagnetic  unit  mag- 
netic pole  strength  is  defined  by  the  equation 


m  =  (47r^2^)*  (3) 

where  L  and  .Fare  expressed  in  c.g.s.  units  and  p  is  expressed 
in  terms  of  the  inductivity  of  free  aether  (/*0)  as  unit  inductivity. 

From  this  fundamental  definition  and  the  general  equations 
already  developed  for  systems  defined  as  rational,  the  definitions 
of  the  rational  electromagnetic  units  of  all  the  other  electrical 
quantities  follow.  Thus  the  REM  unit  magnetic  intensity  is 
defined  by  the  equation  H  =  F/m,  H  being  expressed  in  the 
REM  unit  magnetic  intensity  by  definition,  when  m  is  expressed 
in  the  REM  unit  pole  strength  and  F'm  c.g.s.  units  ;  the  REM 
unit  current  is  defined  by  the  relation  7=  FjBL  or  /=  2ird  x  H\ 
the  REM  unit  electric  charge  from  the  relation  q  =  It;  the  REM 
unit  electric  permittivity  from  the  equation  c  =  $2/4.7rL2F;  etc. 

(2)  The  common  (or  irrational}  system.  If,  without  changing 
the  units  of  inductivity,  force,  or  length,  we  put  A,(i)t  XL,  equal 
to  unity  instead  of  473-,  we  obtain,  as  the  definition  of  the  electro- 
magnetic (EM]  unit  pole  strength,  the  relation 

m  =  (nL2F)*  (4) 

On  comparing  this  equation  with  (3)  we  see  that  the  electro- 
magnetic unit  just  defined  is  equal  to  (471-)*  times  the  electromag- 
netic rational  unit  pole  strength. 


UNITS    AND    DIMENSIONS.  417 

The  system  of  electrical  units  built  up  from  this  unit  pole 
strength  as  fundamental  unit  in  a  manner  exactly  similar  to  that 
in  which  the  rational  system  is  built  up  from  the  REM  \am\.  pole 
strength,  with  the  exceptions  referred  to  in  the  closing  para- 
graph of  §  i  ,  is  called  the  electromagnetic  system  of  units. 

3.  Relations  Between  the  Units  of  Different  Systems.  Every 
equation  developed  in  the  preceding  chapters  holds  good,  as 
already  stated,  by  definition,  when  every  electrical  quantity  oc- 
curring therein  is  expressed  in  its  rational  electrostatic  unit,  or 
when  every  electrical  quantity  is  expressed  in  its  rational  electro- 
magnetic unit,  all  other  quantities  being  expressed  in  c.g.s.  units. 
Every  one  of  these  equations  that  contains  the  definition  of  a 
unit,  moreover,  except  those  defining  unit  charge,  unit  pole 
strength,  and  the  other  units  referred  to  in  the  closing  paragraph 
of  §  I  and  mentioned  in  the  appropriate  places  in  the  text,  is 
valid  also  when  expressed  in  irrational  units  throughout,  either 
all  electrostatic  or  all  electromagnetic.  Thus,  on  any  system 
of  units,  electric  intensity  is  defined  as  the  force  per  unit  charge, 
magnetic  intensity  as  the  force  per  unit  pole  strength,  electric 
displacement  as  permittivity  x  intensity,  capacity  as  charge  per 
unit  voltage,  etc.  The  following  definitional  equations,  in  which 
plain  letters  denote  quantities  expressed  in  rational  units  and 
primed  letters  quantities  expressed  in  irrational  units,  will  serve 
as  examples  :  F  =.  Eq  =  E  q' 

F  =  Hm  =  H'm' 

dr  =  dqjp  =  dqf  jpf 
=Df  jEf 


F/L  =  IB  =  FB 


418          ELEMENTS   OF    ELECTROMAGNETIC    THEORY 


(but  5  =  §DdSI  Fand  S'  =fD'JS/47r  V' 


etc.,  nearly  all  the  equations  being  identical  on  the  two  systems. 
On  the  other  hand,  while  some  of  the  derived  equations  are 
identical  in  the  rational  and  irrational  systems,  many  are  not 
identical.  Thus,  for  example,  on  developing  the  equations  for 
the  irrational  systems,  we  find  that 

IF  =  471-^',  while  II  =  q 

S'  =  TL'/4irV,  '  while  S=U/F 
V  =  q'lcL,  while  F=  q  \qircL 

U'  =    ^/2^     while  U=    cE2 


T!  =  J/x^2/47r,   while  T= 
curl  H'  =  47n7,    while  curl  H  =  i 
while  also 

w=  J5r  F/2  =  |/2/5/  =  %$'  v'  = 
w=  jz'/'2 

curl  E  =  —  dB'jdt,  curl  E=  —  dBjdt      etc.,  etc. 

The  ratio  of  the  rational  electrostatic  unit  of  a  given  quantity 
to  the  irrational  electrostatic  unit  of  the  same  quantity  is  always 
equal  to  the  ratio  of  the  corresponding  rational  electromagnetic 
unit  to  the  irrational  electromagnetic  unit.  This  ratio  for  each 
of  the  principal  electrical  quantities  is  given  in  Table  II. 


UNITS   AND    DIMENSIONS. 


419 


•« 

S 


^  Practical  unit  of  work  =  Joule  *M  ^ 
IT  io7  ergs.  Practical  unit  of  ac-  £ 
6  tivity  =  Watt  =  Joule  per  sec-  £ 
ond=  io7  ergs  per  second.  ^ 

°o 


.1, 


' 


420          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

The  ratio  of  the  rational  electrostatic  unit  to  the  rational 
electromagnetic  unit  of  a  given  quantity  is  always  equal  to  the 
ratio  of  the  irrational  electrostatic  unit  to  the  irrational  electro- 
magnetic unit  of  the  same  quantity.  This  ratio  can  easily  be 
found  in  any  case  from  the  equations  defining  and  connecting  the 
units,  and  the  relation  given  in  §  4,  XII.,  between  the  electro- 
magnetic and  electrostatic  unit  current.  Thus  the  electromag- 
netic unit  charge  or  current  is  a  times  as  great  as  the  corre- 
sponding electrostatic  unit ;  the  electrostatic  unit  magnetic  pole 
strength  is  a  times  as  great  as  the  electromagnetic  unit  pole 
strength  ;  the  electromagnetic  unit  permittivity  is  a2  times  as 
great  as  the  electrostatic  unit  permittivity ;  the  electrostatic  unit 
inductivity  is  a2  times  as  great  as  the  electromagnetic  unit  in- 
ductivity,  etc.,  etc.;  the  ratio  of  the  electrostatic  to  the  electro- 
magnetic unit  being  always  a,  a2,  a~1}  or  a~2. 

The  ratio  of  the  electrostatic  to  the  electromagnetic  unit  of 
each  principal  electrical  quantity  is  given  in  Table  II.  The  table 
also  contains  the  ratio  of  the  rational  electrostatic  unit  of  each 
quantity  to  the  irrational  electromagnetic  unit. 

As  shown  by  the  table,  or  by  the  preceding  statement,  the  per- 
mittivity of  free  sether,  which  is  unity  on  the  electrostatic  systems, 
is  1 1  a2  x  unity  on  the  electromagnetic  systems.  Also,  the  in- 
ductivity of  free  sether,  which  is  unity  on  the  electromagnetic 
systems,  is  I  jo*  x  unity  on  the  electrostatic  systems.  Thus  the 
product  of  the  permittivity  of  free  aether  by  its  inductivity,  both 
measured  on  the  same  system  of  units,  is  equal,  numerically, 
to  I  /a2. 

Of  the  dimensions  (§  6)  of  a  nothing  is  known.  In  all  that 
precedes  we  have  assumed  its  dimensions  to  be  zero,  and  we 
shall  adhere  to  this  assumption  in  what  follows,  that  is  we  shall 
treat  a  as  a  mere  number,  except  where  the  contrary  is  stated. 

4.  Experimental  Determination  of  the  Magnitude  of  a.  To  de- 
termine the  value  of  a  experimentally,  it  is  necessary  only  to  find 
the  ratio  between  the  electrostatic  and  electromagnetic  measures 


UNITS   AND    DIMENSIONS.  421 

of  any  one  electrical  quantity.  This  will  furnish  a  or  a  known 
power  of  a  according  to  what  precedes.  Hence  the  ratio  a  can 
be  determined  in  a  great  variety  of  ways.  For  example,  the 
capacity  of  a  standard  condenser  can  be  determined  in  absolute 
electrostatic  measure  from  the  measurement  of  its  geometrical 
dimensions  (§§1—2,  III.),  and  can  be  determined  in  absolute  elec- 
tromagnetic measure  by  comparison  with  a  resistance  (§60,  XIII.), 
or  a  mutual  inductance  (§58,  XIII.),  or  other  electrical  quantity, 
independently  determined  in  absolute  measure  according  to  the 
methods  described  above,  or  other  methods.  By  such  methods, 
and  a  number  of  others,  some  of  which  will  be  apparent  from 
the  methods  of  measurement  previously  discussed  in  this  book, 
a  has  been  determined  with  considerable  precision.  The  best 
results  all  lie  close  (within  a  few  tenths  per  cent.)  to  3  x  io10. 

For  some  of  the  most  recent  and  best  determinations,  see 
Thomson  and  Searle,  Phil.  Trans.,  A,  1890;  H.  Abraham,  Ann. 
de  Chim.  et  de  Phys.,  Vol.  27,  1892  ;  and  D.  Hermuzescu,  Ann. 
de  Chim.  et  de  Phys.,  Vol.  io,  1897. 

According  to  the  theory  developed  in  Chapter  XVI.,  electro- 
magnetic waves  are  propagated  in  free  aether  with  the  velocity 
v=  I/fa,/*,,)*,  which  is  equal  to  a  numerically.  This  velocity  has 
been  determined  approximately  for  long  waves,  and  very  accu- 
rately for  extremely  short  waves  (light),  and  found  to  agree  with 
a  as  otherwise  determined  within  the  limits  of  error  of  experiment. 

5.  Practical  Units.  The  rational  systems  of  units  are  at  present 
unfortunately  but  little  used,  and  the  irrational  electrostatic  system 
is  used  only  in  pure  science  and  mostly  for  theoretical  purposes. 
The  irrational  electromagnetic  system,  on  the  other  hand,  is  more 
extensively  used.  For  the  purposes  of  ordinary  experimental 
work,  however,  especially  for  the  purposes  of  electrical  engi- 
neering, many  of  the  electromagnetic  units,  as  well  as  many  of 
the  c.g.s.  mechanical  units,  are  inconveniently  large  or  small. 
Hence  a  practical  system  of  units,  each  a  decimal  multiple  or 
submultiple  of  the  corresponding  (irrational)  electromagnetic  unit, 


422    ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

or  the  c.g.s.  mechanical  unit,  has  been  developed  in  which  each 
of  the  units  most  frequently  employed  in  practice  has  a  magni- 
tude of  the  same  order  as  that  of  the  quantities  with  which  it  is 
ordinarily  to  be  compared.  These  units  are  so  chosen  as  to  form 
a  self-consistent  system  satisfying  all  the  equations  satisfied  by 
the  electromagnetic  system,  except  those  equations  which  are  not 
wholly  made  up  of  quantities  whose  units  are  defined  in  the  prac- 
tical system.  The  units  of  mass,  length,  permittivity,  and  indue* 
tivity  are  the  same  in  both  systems.  The  relations  between  the 
other  practical  units  and  the  electromagnetic  units  are  given  in 
Table  II.  together  with  the  names  of  the  practical  units. 

A  unit  one  million  (iO6)  times  as  great  as  any  one  of  these 
units  is  designated  by  the  name  of  the  unit  with  the  prefix  mega 
or  meg.  Thus  a  megohm  is  one  million  ohms. 

A  unit  one  millionth  (io~G)  as  great  as  any  one  of  these  units 
is  designated  by  the  name  of  the  unit  with  the  prefix  micro. 
Thus  a  microvolt  or  microfarad  is  one  millionth  of  a  volt  or  a 
farad. 

In  like  manner  the  prefixes  deka,  deci,  centi,  etc.,  are  attached 
to  the  names  of  the  units  with  the  same  effects  as  they  have  upon 
the  common  units  of  the  metric  system. 

6.  The  Dimensions  of  Electrical  Quantities.*  As  already  stated, 
nothing  is  known  of  the  physical  nature,  or  dimensions  in  mass, 
length,  and  time,  of  the  quantities  c,  JJL,  and  a.  Hence,  since 

*  Vector  Dimensions.  In  the  system  of  dimensions  adopted  here  no  account  has 
been  taken  of  the  fact  that  a  length,  unlike  a  mass  or  a  time,  is  a  vector,  or  directed 
quantity.  Thus,  on  this  system,  the  dimensions  of  a  plane  angle,  which  is  a  length 
divided  by  a  length,  are  zero  in  [M"\,  [£],  and  [71],  although  one  of  the  lengths  is 
perpendicular  to  the  other  ;  the  dimensions  of  a  solid  angle  are  zero,  although  it  is  a 
surface  divided  by  the  square  of  a  length  perpendicular  thereto ;  the  dimensions  of 
work  ( force  X  distance  in  direction  of  force)  are  equal  to  the  dimensions  of  torque 
(force  X  distance  perpendicular  to  force),  etc. ,  etc.  Yet  there  is  an  essential  difference 
between  a  mere  number,  which  of  course  has  no  dimensions,  and  a  plane  or  solid 
angle,  and  there  is  an  essential  difference  between  the  physical  nature  of  a  quantity  of 
work  and  a  torque. 

These  anomalies  vanish,  however,  if  we  express  all  dimensions  in  terms  of  [J/], 
[71],  and  \_X~\,  [  V],  and  [Z],  three  lengths  at  right  angles  to  one  another,  thus 
taking  account  of  the  vector  nature  of  L.  On  this  system,  a  plane  angle  has  such 


UNITS   AND    DIMENSIONS.  .     423 

every  electrical  unit  involves  one  or  more  of  these  quantities,  the 
complete  dimensions  of  every  electrical  quantity  are  unknown. 
If,  however,  [V],  [/A],  and  \a~\  are  written  for  the  unknown 
dimensions  in  mass,  length,  and  time  of  c,  /A,  and  a,  respectively, 
a  complete  expression  for  the  dimensions  of  every  electrical  quan- 
tity can  be  written  down.  Thus,  if  the  dimensions  of  any  quan- 
tity are  obtained  from  the  equations  of  either  of  the  electrostatic 
systems,  they  will  be  expressed  in  terms  of  [-#/],  \L\  ,  \_T~\,  and 
[a\  and  \_c\  ;  if  they  are  obtained  from  the  equations  of  either 
electromagnetic  system,  they  will  be  expressed  in  terms  of  [  J/]  , 
[Z,],  [T~\,  [#],  and  [/A].  The  dimensions  of  all  the  principal 
electrical  quantities,  both  in  terms  of  [a]  and  [c]  and  in  terms 
of  [#]  and  [ft],  are  given  in  Table  II. 

Since  the  actual  dimensions  in  \_M~\,  [£],  and  [T~\  of  any 
electrical  quantity  must  be  the  same  whether  expressed  in  terms 
of  \a~\  and  [r]  or  in  terms  of  [a\  and  [/A],  the  dimensions  of 
any  quantity  in  terms  of  \_a]  and  [r]  may  be  equated  to  its  di- 
mensions in  terms  of  \_a\  and  [/A]  .  Thus,  equating  the  dimen- 
sions of  electric  charge  in  terms  of  [<?]  and  [V]  to  its  dimensions 
in  terms  of  [a]  and  [/A],  we  obtain 


[a/W]       [Z/7-]  (5) 

That  is,  the  quantity  a/c*fri  is  a  linear  velocity.  Exactly  the  same 
relation,  (5),  and  only  this  relation,  follows  from  equating  the  two 
expressions  for  the  dimensions  of  any  other  electrical  quantity. 

The  velocity  of  electromagnetic  waves  as  determined  from  the 
equations  of  Chapter  XVI.  is  I  /(/JLC)*.  In  these  equations,  how- 
ever, as  stated  above,  the  dimensions  of  a  are  ignored.  If  this 
is  not  done,  it  is  easy  to  see  that  the  velocity  comes  out  equal 
to  i/(/Lur)*x  [#],  which  is  numerically  equal  to  i/(V/A)*  and  is 
dimensionally  correct  by  (5). 

dimensions  as  [Jfl^-1]  or  [YZ—l~\;  a  solid  angle  such  dimensions  as  [XYZ~2~\  or 
(  FZJf-2);  a  quantity  of  work  such  dimensions  as  \MX'2T—r\\  a  torque  such  dimen- 
sions as  \MXYT~^~\\  etc.  This  is  therefore  a  rational  system  of  dimensions.  This 
system  of  dimensions,  as  applied  to  mechanical  and  electrical  quantities,  is  discussed 
at  length  by  W.  Williams,  in  the  Philosophical  Magazine,  September,  1892. 


CHAPTER   XV. 

THE   GENERAL   ELECTRIC   CURRENT. 

1.  Displacement  Current  and  Magnetic  Intensity.  It  has  been 
shown  in  §  27,  XIII.  ,  that  when  no  other  kind  of  current  is 
present  the  conduction  current  across  a  surface  is  equal  to  the 
m.m.f.  around  the  edge  of  the  surface,  and  that  the  conduction 
current  density  is  equal  to  the  curl  of  the  magnetic  intensity. 

That  a  changing  electric  displacement,  or  a  pure  displacement 
current,  also  gives  rise  to  a  magnetic  field  similar,  qualitatively, 
to  that  of  a  conduction  current  Hertz  proved  by  direct  experi- 
ments. Consistently  with  these  experiments,  we  shall  here  as- 
sume that  a  given  displacement  current  develops  a  magnetic 
field  similar,  both  qualitatively  and  quantitatively,  to  that  con- 
nected with  a  conduction  current  of  the  same  magnitude  and 
distribution.  The  very  important  consequences  of  this  assump- 
tion are  in  rigorous  accord  with  experiment  (XVI.).  Thus  we 
may  write  for  a  closed  curve  through  which  the  electric  flux  is 
changing  and  through  which  there  is  no  other  form  of  current 
than  a  displacement  current, 


(i) 

and  curl  H=id  =  dDjdt  (2) 

which  are  analogous  to  (8)  and  (9),  XIII. 

fl  is  called  an  induced  m.m.f.  ,  and  H  an  induced  magnetic 
intensity. 

2.  The  Magnetic  Field  Induced  by  the  Motion  of  a  Concentrated 
Charge.      Let   an   approximately  concentrated    charge  q  =  pdr 

424 


THE   GENERAL    ELECTRIC    CURRENT. 


425 


move  with  a  velocity  v,  Fig.  130.  Let  II  denote  the  electric 
flux  through  a  circle  of  any  radius  a  with  its  axis  passing  through 
dr  parallel  to  v,  and  let  the  direction  of  v  be  chosen  as  the  posi- 
,tive  direction  through  the  circle.  Owing  to  the  motion  of  the 
charge  with  its  field  the  flux  through  the  circle  will  increase  at 
the  rate  dKjdt  and  a  m.m.f.  equal  to  fl  =  dtt/dtwi\\  be  induced 
in  the  positive  direction  around  the  circle.  By  symmetry,  the 


Fig.  130. 

induced  magnetic  intensity  His  equal  in  magnitude  at  all  points 
•of  the  circle  which,  like  all  other  parallel  circles  centered  on  the 
same  axis,  is  a  line  of  magnetic  intensity.  Let  0  denote  the 
angle  between  v  and  the  direction  of  D  at  every  point  of  the  cir- 
cle ;  then 

dTLjdt  =  v  •  sin  6  •  2TraD  =  fl  =  2iraH 

and  the  directions  of  the  vectors  are  so  related  that  this  equation 
gives 

H=\/vD>smO  (3) 

which  is  analogous  to  (50),  XIII. 

Thus  the  magnetic  intensity  is  developed  by  the  motion  of  the 
tubes  of  electric  displacement  at  right  angles  to  their  length. 

Since  in  the  case  considered 


D  = 


426         ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

where  r_  (•=  r  numerically)  is  the  distance  of  the  circle  from  dr, 
(3)  is  equivalent  to 

H  =  I  /4?rr3  •  Vvq  •  r_  sin  6  =  drj^rrr2  •  Vpv  -rsinO 

r  sin  0     '4'' 


Thus,  as  shown  by  a  comparison  of  (4)  with  (13),  XII.  ,  the 
same  magnetic  effects  are  produced  by  a  moving  charge  with 
convection  current  density  ta  as  by  a  conduction  current  of  den- 
sity te  =  icv. 

In  strictness  the  above  results  are  only  approximate  and  A  re- 
quire an  appreciable  correction  when  v  is  (^niparablc-with  the 
velocity  of  free  electromagnetic  disturbances  (the  velocity  of 
light),  §  II,  XVI.,  since  the  field  at  a  distance  from  a  moving 
charge  lags  behind  the  charge. 

A  similar  magnetic  field  would  be  developed  by  the  motion 
of  the  pole  of  an  electret,  and  an  electric  field  exactly  analogous 
to  this  last  by  the  motion  of  the  pole  of  a  magnet,  only  the 
space  outside  the  magnet  or  electret  being  considered. 

3.  The  Magnetic  Field  of  a  Cylindrical  Convection  Current. 
The  electric  field  of  a  cylindrical  condenser  is  discussed  in  §  §  8-9, 
II.  Let  the  charge  upon  unit  length  of  the  inner  conductor  be 
-f  q  and  that  upon  unit  length  of  the  outer  conductor  therefore 
—  q  ;  and,  for  simplicity,  suppose  the  conductivity  ot  the  outer 
cylinder  perfect.  Then  the  tubes  of  displacement  will  terminate 
normally  upon  the  outer  cylinder  whether  the  inner  cylinder  is 
at  rest  or  in  motion.  Let  the  inner  cylinder  move  with  a  velocity 
v  in  the  direction  AB  of  its  axis.  Then  the  convection  current 
through  any  closed  curve  surrounding  the  inner  cylinder  is 

L  =  & 

in  the  direction  AB. 

By  the  last  article,  the  m.m.f.  around  any  closed  curve  sur- 
rounding the  inner  cylinder  and  within  the  outer  cylinder  is 

therefore 

XI  =  7= 


THE   GENERAL    ELECTRIC    CURRENT.  427 

The  magnetic  intensity  has  the  same  magnitude  at  all  points 
of  any  circle  of  radius  r  coaxial  with  the  cylinders,  and  all  such 
circles  are  lines  of  magnetic  intensity.  Hence 

fl  ==  27rrH '  =  qv  =  2TrrDv      ^ 
and,  with  due  respect  to  the  directions  of  the  vectors, 

H=VvD  (5) 

In  this  case  there  is  no  displacement  current  through  the  cir- 
cle of  intensity,  but  the  magnetic  field  is  developed  as  before  by 
the  motion  of  tubes  of  displacement  perpendicularly  to  their 
length.  (5)  is  a  particular  case  of  (3),  since  in  this  case  sin  6 
=  i. 

4.  Experiments  upon  the  Convection  Current.  That  an  electric 
convection  current,  in  conformity  with  the  above  theory,  is  accom- 
panied by  a  magnetic  field  of  the  same  character,  both  qualita- 
tively and  quantitatively,  as  that  connected  with  a  pure  conduc- 
tion current  of  the  same  magnitude  and  distribution,  has  been 
proved  in  several  series  of  experiments  by  Rowland,  using  charged 
rotating  discs,  and  has  been  confirmed  by  many  others. 

Just  as  the  magnetic  field  of  a  conduction  current  may  be 
deduced  as  a  consequence  of  Ampere's  law,  §  n,  XII.,  so,  con- 
versely, Ampere's  law  may  be  deduced  as  a  consequence  of  the 
(experimentally  investigated)  magnetic  field  of  the  current.  Since 
therefore  a  convection  current  develops  a  magnetic  field  identical 
with  that  of  a  conduction  current  of  the  same  magnitude  and 
distribution,  Ampere's  law  must  apply  to  such  a  current.  Hence 
a  beam  of  cathode  rays  (consisting  of  very  fine  negatively  charged 
particles,  or  electrons,  moving  with  velocities  approaching  that  of 
light)  should  be  deflected  when  immersed  in  a  magnetic  field 
perpendicular  to  the  beam.  That  such  a  deflection  occurs,  in 
qualitative  agreement  with  the  theory,  has  been  known  for  many 
years.  A  deflection  in  a  magnetic  field,  in  the  direction  indicated 
by  theory,  of  the  much  more  massive  and  more  slowly  moving 
positively  charged  particles  forming  the  canal  rays  has  also  been 


428         ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

recently  observed.  The  assumption,  justified  by  the  experiments 
of  Rowland,  that  the  agreement  between  these  experiments  and 
theory  is  also  quantitative  has  recently  led  to  very  important 
advances  in  knowledge. 

In  accordance  with  what  precedes,  we  may  write  for  the  m.m.f. 
around  the  edge  of  a  surface  across  which  there  is  a  convection 

current  only  0        r  >a% 

11  =  yc»  (6) 

and  curl  H  =  icv  (7) 

5.  Magnetic  Intensity  and  the  Motion  of  Electric  Displacement. 
Motional  Magnetic  Intensity.  In  §  2  it  has  been  shown  that  for 
the  case  there  considered  (i),  and  therefore  (2),  is  equivalent  to 
(3).  In  the  same  way,  just  as  (8)  and  (9),  XIII. ,  are  equivalent 
to  (5#),  XIII. ,  so  (i)  and  (2)  may  be  readily  shown  to  be 
equivalent,  in  the  general  case,  to  the  more  general  relation 

H  =  VvD  sin  d  (8) 

where  H  is  the  magnetic  intensity  at  a  point  P  in  the  dielectric, 
at  which  the  displacement  is  D,  developed  by  the  motion  of  the 
tubes  of  displacement  relatively  to  the  medium  at  P  with  the 
velocity  v,  whose  component  perpendicular  to  D  is  v  sin  6. 

If  the  medium  at  P  is  in  actual  motion  with  reference  to  the 
surrounding  medium,  containing  the  inducing  system,  such  as 
fixed  charges,  it  is  an  intrinsic  magnetic  intensity,  and  is  called  a 
motional  magnetic  intensity.  The  existence  of  a  motional  mag- 
netic intensity  was  first  established  by  Roentgen,  in  whose  ex- 
periments a  magnetic  field  in  qualitative  agreement  with  (8),  to 
judge  from  its  continuation  in  the  air,  was  developed  in  a  slab  of 
rigid  dielectric  rotated  in  air  between  fixed  charged  discs.  For 
the  most  recent  experiments  upon  the  subject,  see  A.  Eichen- 
wald,  Ann.  der  Physik,  Nos.  5  and  6,  1903.  The  experiments 
indicate  that  for  D,  in  (8),  if  c2  and  cl  (=  CQ,  sensibly)  denote  the 
permittivities  of  the  slab  and  the  air,  respectively, 


THE   GENERAL    ELECTRIC    CURRENT.  429 

should  be  substituted.  That  is,  a  fictitious  convection  current 
produces  the  same  magnetic  effects  as  a  true  convection  current 
of  the  same  magnitude  and  distribution. 

Thus  magnetic  intensity  is  induced  by  moving  tubes  of  electric 
displacement,  as  electric  intensity  is  induced  by  moving  tubes  of 
magnetic  induction  (§  6,  XIII.).  It  cannot  be  said,  however, 
that  magnetic  intensity  is  always  due  to  moving  tubes  of  electric 
displacement,  or  that  electric  intensity  is  always  due  to  moving 
tubes  of  magnetic  induction,  an  attempt  to  make  ($#),  XIII. ,  and 
(8)  general  expressions  for  magnetic  and  electric  intensity  lead- 
ing to  apparently  insurmountable  difficulties.  It  is  sufficient  to 
mention  the  field  of  a  static  electric  charge  or  magnetic  pole. 

6.  (Induced  ?)  Intensities  in  the  Field  of  a  Steady  Conduction 
Current.  Let  the  current  /  traverse  the  inner  and  outer  con- 
ductors of  a  cylindrical  condenser  axially  in  opposite  directions, 
the  inner  cylinder  in  the  direction  AB,  and  suppose  both 
cylinders  perfect  conductors.  Then  the  electric  field  will  be 
radial  and  perpendicular  to  both  cylinders  as  in  a  purely  static 
field. 

Imagine  the  conduction  current  to  consist  in  the  motion  of  the 
positive  and  negative  ends  of  the  electric  tubes  along  AB  and 
CC  respectively,  the  electric  tubes  travelling  in  the  direction  AB 
with  the  velocity  v.  Then  the  magnetomotive  force  around  a 
circle  of  magnetic  intensity  of  radius  r  is 

fl  =  /  =  2-rrrH '=  qv  =  2TrrDv 

q  being  the  charge  upon  unit  length  of  the  inner  cylinder  and  D 
the  displacement  at  the  distance  r  from  the  axis.  With  due  re- 
spect to  the  signs  of  the  vectors,  the  last  equation  gives 

H=  VvD  (9) 

Thus  by  assuming  that  the  magnetic  field  of  the  steady  con- 
duction current  is  a  consequence  of  the  (assumed)  motion  of  its 
electric  field  we  are  led  to  the  same  relation  between  Ht  v  and  D 
as  already  deduced  for  the  displacement  current. 


43°          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

Jf  we  imagine  the  electric  field,  likewise,  to  be  developed  by 
the  (assumed}  motion  of  the  magnetic  field  with  velocity  u,  (50), 

XIII.,  gives 

E=VBu  (10) 

Equations  (9)  and  (10)  give  for  the  relations  between  u  and  v 

UV  =    I  fffl 

ulv=  \EDl\HB 
u\HB  =  v\ED  =  \EH 

Since  the  electric  and  magnetic  energy  densities  are  not  in 
general  equal,  except  in  the  case  of  pure  electromagnetic  waves 
(XVI.),  the  above  conceptions  lead  to  the  anomalous  result  that 
u  and  v  are  different  and  may  have  any  ratio  to  one  another. 
Since  in  some  cases  (according  to  the  dissociation  theory),  and 
probably  in  all  (§  1  5,  IX.),  the  steady  electric  current  in  an  actual 
conductor  consists  in  the  motion  throughout  the  conductor  in 
opposite  directions  of  positively  and  negatively  charged  particles, 
and  since  the  surface  charges  connected  with  the  external  field 
of  the  steady  current  do  not  take  part  in  the  conduction  (§9, 
VIII.),  the  above  results  must  be  taken  as  at  present  only  sug- 
gestive. 

7.  The  First  Law  of  Circuitation  for  Media  at  Rest  and  in  Mo- 
tion.  When  a  given  surface  in  a  medium  at  rest  is  crossed  by 
conduction,  convection,  and  displacement  currents  simultaneously, 
the  total  current  /  through  the  surface  and  the  m.m.f.  fl  around 
its  edge  (the  direction  of  the  m.m.f.  being  related  to  that  of  the 
current  as  the  direction  of  rotation  to  that  of  translation  of  a 
right-handed  screw)  are  connected  by  the  equation 


and  the  total  current  density  i  and  the  magnetic  intensity  H  are 
connected  by  the  equation 


THE   GENERAL    ELECTRIC    CURRENT.  431 

where  i  -\-  ij  -4-  i    is  a  vector  sum.      Here  H  denotes  either  the 

c     '       a     '       cv 

intensity  due  to  the  currents,  or  the  total  intensity  due  to  both 
currents  and  magnets,  if  such  are  present  (since  the  curl  of  the 
intensity  due  to  the  poles  of  a  magnet  is  zero). 

If  the  medium  is  in  motion,  with  the  velocity  —  v  at  the  point 
where  the  electric  displacement  is  D  (§5),  and  if  //still  denotes 
the  total  magnetic  intensity,  the  curl  of  the  motional  magnetic 
intensity,  which  we  shall  call  the  motional  current  density,  or 
the  fictitious  convection  current  density,  and  denote  by  im,  must  be 
added  to  the  first  member  of  (12).  Thus  we  have, 

i  =  *,  +  *„  +  C  +  C=  curl  H  (x 3) 

the  first  member  being  a  vector  sum.  This  is  the  most  general 
form  of  the  first  law  of  circulation. 

8.  The  Circuital  Character  of  the  Total  Current.  Kirchhoff's 
Law  I.  Generalised.  That  a  steady  conduction  current  flows  in 
a  closed  circuit  (div  ic  =  o  everywhere)  is  shown  in  §5,  VIII. 

From  the  Cartesian  expression  for  the  divergence  of  a  vector 
(§31,  I.),  and  the  Cartesian  expression  for  the  curl  of  a  vector 
(§4,  XVI.),  it  follows  that  the  divergence  of  the  curl  of  any 
vector  is  zero.  Hence  it  follows. from  (12)  and  (13),  since  z,  the 
total  current  density,  is  equal  to  curl  //,  that 

div  i  =  div  curl  H  —  o  (14) 

That  is,  in  any  case,  the  total  current  flows  in  closed  tubes. 

Thus,  for  example,  let  an  electric  condenser  AB  be  discharged 
through  a  wire  C,  and  first  suppose  the  capacity  of  the  wire 
negligible  in  comparison  with  that  of  the  rest  of  the  system. 
Then  during  the  discharge  the  conduction  current  /=  —dqjdt 
will  be  sensibly  the  same  across  every  section  of  the  wire  (if  the 
capacity  of  the  wire  were  zero,  any  charge  there  accumulating 
would  produce  an  infinite  potential  difference),  and  the  displace- 
ment and  displacement  current  will  be  confined  sensibly  to  the 
region  occupied  by  the  tubes  stretching  from  A  to  B.  During 


432  ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

the  discharge  the  flux  from  A  to  B  decreases  at  the  rate  —  dqjdt 
and  the  flux  from  B  to  A  increases  at  the  same  rate  —  dqjdt. 
Thus  the  displacement  current  dH.fdt  =  —  dqjdt  from  B  to  A 
through  the  dielectric  across  any  closed  surface  surrounding  one 
of  the  plates  is  equal  to  the  conduction  current  across  every  sec- 
tion of  the  wire  from  A  to  B.  Thus  the  total  current  flows  in  a 
closed  circuit,  the  displacement  current  starting  where  the  con- 
duction current  stops. 

Similar  phenomena  occur  when  the  wire  is  cut  and  its  ends 
connected  to  the  terminals  of  a  voltaic  cell.  The  conduction 
current  through  the  wire  and  cell  as  the  condenser  is  charged  is 
equal  to  the  displacement  current  in  the  same  direction  around 
the  circuit  through  the  dielectric. 

If  the  capacity  of  the  wire  C  differs  from  zero  (which  is  always 
the  case  to  a  greater  or  less  extent),  then,  if  the  condenser  is  dis- 
charged by  bringing  the  wire  into  contact  with  the  plates  at  A 
and  B  simultaneously,  tubes  of  displacement  will  move  out  along 
the  wire  stretching  from  the  part  near  A  to  the  part  near  B,  giving 
rise  to  a  displacement  current  through  the  dielectric  from  the 
one  part  to  the  other  as  well  as  to  a  conduction  current  through 
the  wire ;  and  the  sum  of  the  two  currents,  through  the  wire  and 
through  the  external  dielectric  from  A  to  B,  is  equal  to  the  dis- 
placement current  from  B  to  A  through  the  rest  of  the  dielectric. 
During  this  process  the  conduction  current  is  not  constant  from 
section  to  section  of  the  wire,  being  zero  across  the  more  remote 
parts  of  the  wire  while  it  has  an  appreciable  value  across  the 
nearer  parts  immediately  after  the  beginning  of  the  disqharge. 


CHAPTER    XVI. 

THE   TRANSFERENCE   OF    ELECTROMAGNETIC    ENERGY. 
ELECTROMAGNETIC   WAVES.     MAXWELL'S    THEORY. 

1.  Poynting' s  Theorem  *  when  E  is  Perpendicular  to  H.  Fig. 
131  represents  one  end  of  a  system  consisting  of  two  long  coaxial 
perfectly  conducting  circular  cylinders  A  and  Ft  with  external 
and  internal  radii  Rl  and  R2  respectively,  closed  by  a  non-con- 
ducting slab  E  of  zero  permittivity,  and  plugged  with  a  closely 
fitting  right  cylindrical  conductor  D  of  length  L  and  resistance 


\   ' 

-R- 

->> 

| 

m* 

A 

A 

-*- 

[V  1 

5 

c 

**"• 

0»  ~    , 

W: 

M 

Fig.  131. 

IV.  A  and  /^  are  connected  at  the  remote  end  of  the  system 
with  the  positive  and  negative  terminals  of  a  voltaic  cell  or  other 
electric  generator,  and  are  traversed  by  a  constant  current  in  the 
direction  ABCDFF. 

Since  A  and  F  are  perfect  conductors  and  since  E  has  zero 
permittivity,  the  electric  and  magnetic  fields  are  confined  wholly 
to  the  dielectric  and  to  the  conductor  D.  (If  the  permittivity  of 
E  were  not  zero,  the  fields  would  be  the  same  except  that  in 
addition  a  static  electric  field  would  extend  beyond  C  to  the  right 
connecting  F  and  C.)  Also,  the  lines  of  electric  intensity  are 

*J.  H.  Poynting,  Phil.  Trans.,  Part  II.,  1884  and  Part  II.,  1885  ;  Oliver 
Heaviside,  Electrical  Papers  and  Electromagnetic  Theory ;  J.  H.  Poynting,  Rap- 
ports, Congres  Int.  de  Physique,  1900,  Vol.  II. 

433 


434          ELEMENTS    OF    ELECTROMAGNETIC   THEORY. 

normal  to  both  cylinders  and  straight,  as  in  a  purely  static  field. 
The  lines  of  magnetic  intensity  are  circles  centered  on  the  axis 
of  the  cylinders  in  planes  perpendicular  thereto. 

Let  the  steady  current  be  denoted  by  /,  and  the  steady  differ- 
ence of  potential  from  A  to  F  (there  is  no  fall  of  potential  along 
A  or  along  F)  by  V=  WI. 

Energy,  generated  by  the  voltaic  cell  (or  other  generator),  is 
supplied  to  the  system  ABF  at  the  rate  VI,  and  is  dissipated  by 
the  resistance  Wat  the  rate  WP  =  VL  Since  there  is  no  energy 
within  the  conductors  A  and  F  or  outside  the  closed  system, 
energy  must  therefore  be  transferred  in  the  direction  ABC  across 
every  section  of  the  dielectric  at  the  rate 


f  (i) 

Since  V=jEdr  along  a  line  of  electric  intensity,  and  /=  2'jrrH, 
where  E  and  H  denote  the  electric  and  magnetic  intensities  at  a 
circle  in  the  dielectric  distant  r  from  the  axis,  (  I  )  may  be  written 


(R)  =  VI  =EH-  2irrdr  = 

where  6"  denotes  the  cross-section  feirrdr  of  the  dielectric.  There- 
fore, due  attention  being  paid  to  the  signs  of  the  vectors,  the  time 
rate  per  unit  area,  R,  at  which  electromagnetic  energy  is  trans- 
ferred across  an  area  whose  plane  contains  E  and  Ht  perpendic- 
ular to  one  another,  or  the  electromagnetic  energy  flux  density  ',  is 

R  =  d(R)jdS=VEH  (2) 

This  is  Poynting's  theorem  for  the  case  in  which  E  is  perpen- 
dicular to  H.  A  more  general  form  of  the  theorem  will  be  de- 
veloped in  §  5. 

Let  L  and  6*  denote  the  inductance  and  capacity  of  unit  length 
of  the  system,  then  \LP  and  JSF2  denote  the  magnetic  and 
electric  energies  contained  in  a  unit  length  of  the  dielectric.  Let 
:-  denote  the  velocity  with  which  the  electric  energy  moves  in 
the  direction  AB,  and  u  the  velocity  with  which  the  magnetic 


THE   FLUX   OF    ELECTROMAGNETIC    ENERGY. 


435 


energy  moves  in  the  same  direction  (see  §  6,  XV.).     Then  we 

may  write 

(R)  =  HJSF2)  +  uQ£J*)  (3) 

which  is  equal  to  VI  if 

uv  =  I  JSL  =  i  /cfji  (see  below)  (4) 

In  exactly  the  same  way  we  have  for  the  energy  flux  density 

R  =  v(\c&)  +  u(lnH*)  (5) 

which  is  equal  to  EH  if  uv  =  I  jc^. 

Thus  the  conception  of  moving  tubes  is  consistent  with  Poynt- 
ing's  theorem  if  the  relation  (4)  holds  between  the  velocities. 

When    Z/2  =    SF2  or 


'(6) 

2.  Mechanical  Analogue.  Consider  a  circular  cylindrical  rod 
rotating  uniformly  about  its  axis  AB  and  transmitting  power  in 
the  direction  AB.  Let  the  constant  angular  velocity  be  denoted 
by  /and  the  torque  acting  across  eveiy  section  by  Vy  the  com- 
mon direction  of  both  being  related  to  the  direction  AB  as  the 
direction  of  rotation  to  that  of  translation  of  a  right-handed 
screw. 

The  rate  at  which  energy  is  transferred  across  every  section 
of  the  rod  in  the  direction  AB  is 


dr 


Owing  to  the  torque  across  every  section,  the  rod  is  twisted, 
or  any  two  sections  are  sheared  with  respect  to  one  another. 

f*& 

(g 

Fig.  132. 

For  all  points  at  a  given  distance  r  from  the  axis  the  (angle  of) 
shear,  or  relative  shift  per  unit  length  between  two  cross-sections, 


A  

- 

)  ' 

* 

II-2-—  B 

436          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

is  the  same  and  will  be  denoted  by  D.  Fig.  132  shows  the 
natural  and  sheared  states  of  a  ring  of  radius  r  and  infinitesimal 
thickness  dr.  D  is  zero  at  the  axis  and  is  proportional  to  ry 
having  its  greatest  value  at  the  surface  of  the  rod.  The  direction 
of  D  is  everywhere  radial  from  the  axis,  since  a  right-handed 
screw  at  any  point  rotating  in  the  direction  of  the  twist  or  shear 
at  the  point,  as  shown  by  the  arrows  in  the  figure,  would  move 
radially  toward  the  surface. 

Let  the  modulus  of  rigidity  of  the  rod  be  denoted  by  ;/,  and  its 
reciprocal,  or  the  shear  permittivity,  by  c.  Then,  at  any  point 
of  the  ring  considered,  the  shearing  stress  (shearing  force  per 
unit  area,  or  shearing  torque  per  unit  volume),  which  will  be 
denoted  by  E,  has  the  same  direction  as  that  of  D  and  is  equal  to 

E=nD  =  Djc 

Since  the  area  of  the  cross-section  of  a  ring  of  radius  r  and 
thickness  dr  is  2'irrdr,  the  torque  about  the  axis  acting  upon  the 
cross-section  is 

dV—  27rrdrEr 

Let  the  linear  velocity  at  any  point  distant  r  from  the  axis  be 
denoted  by  H.  Then 

H=rl 

The  rate  at  which  energy  is  transferred  across  the  zone  of 
radius  r  and  width  dr  is 


IdV=  2Trr2drEHjr 
and  the  rate  of  transfer  per  unit  area  is 

R=VEH  (a) 

if  due  attention  is  paid  to  signs. 

The  potential  energy  per  unit  volume  is  ^cE2  =  %nD2. 

Let  the  density  of  the  rod  be  denoted  by  ft.     Then  the  kinetic 
energy  per  unit  volume  is  J/^//2. 

If  we  assume  the  potential  energy  to  travel  in  the  direction  AB 


THE    FLUX   OF    ELECTROMAGNETIC    ENERGY.          437 


with  the  velocity  v  and  the  kinetic  energy  with  the  velocity  u, 

we  have  also 

R  = 


which  is  consistent  with  (a)  if  uv  =  i  /CJJL.     Ifu  =  v 

R  =  v($c&  + 
which,  combined  with  (a),  gives 

v  =  MEHj^cE2  + 


3.  Two  Perfectly  Conducting  Parallel  Circular  Plates  Connected 
by  a  Right  Circular  Coaxial  Conducting  Cylinder.  Let  Z,  Fig. 
133,  denote  the  distance  between  the  plates,  or  the  length  of  the 


I    B 


cylinder,  a  the  radius  of  the  cylinder,  both  supposed  small  in 
comparison  with  the  radius  of  the  plates,  and  IV  the  resistance 
of  the  cylinder.  Let  the  plates  be  maintained  at  the  constant 
difference  of  potential  V. 

The  electric  field  between  the  plates  (except  near  their  edges) 
is  uniform  and  parallel  to  the  axis  of  the  cylinder  in  the  direction 
AB.  The  electric  intensity  is  E  =  VjL(=  Djc  in  the  dielectric). 


43$          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

The  current  flows  in  radial  stream-lines  in  the  plane  A  A  toward 
the  axis  of  the  cylinder,  through  the  cylinder  parallel  to  its  axis 
in  the  direction  ABy  and  from  the  cylinder  in  radial  stream  lines 
in  the  plane  BB. 

The  lines  of  magnetic  intensity  are  circles  parallel  to  the  planes 
and  centered  on  the  axis  of  the  cylinder.  The  intensity  at  all 
points  of  a  circle  of  radius  r  is  H  =  7/27rr  =  VJ27rrW,  if  r  is 
greater  than  a.  If  r  is  less  than  a,  H  =  rIJ27ra2. 

Within  the  perfect  conductors  there  is  no  field  of  either  kind. 

Energy  is  dissipated  in  the  cylinder  at  the  rate  (R)  =  WP  — 
VI.  Hence  energy  is  transferred  inwardly  across  every  cylin- 
drical surface  with  its  ends  in  the  planes  A  and  B  and  enclosing 
the  cylinder  at  the  same  rate 

(R)  =  VI  (a) 

If  the  surface  is  a  right  circular  cylinder  of  radius  r  coaxial 
with  the  conducting  cylinder,  (a)  becomes 

(R)  =  EL  x  27rr  x  H=  EH-  2-rrrL 

But  2irrL  is  the   area  of  the   surface   considered.      Hence   the 
energy  flux  density  is,  if  due  attention  is  paid  to  signs, 

R=MEH  (6) 

as  already  proved  for  perpendicular  intensities  in  §  i. 

The  energy  contained  in  each  tube  of  displacement  per  unit 
cross-section  is  \  VD.  The  velocity  of  the  tubes  of  displacement 
(see  §  6,  XV.),  or  of  the  electric  energy,  inward  at  the  distance  r 
(greater  than  a)  from  the  axis,  if  we  assume  that  (8),  §  5,  XV., 
applies  to  the  field  of  the  steady  conduction  current,  is 

(c) 


Hence  the  rate  at  which  electric  energy  crosses  the  surface 
inward,  or  the  rate  at  which  electric  energy  is  dissipated  in  the 
conductor,  is,  according  to  the  conception  of  moving  tubes, 


THE    FLUX   OF    ELECTROMAGNETIC    ENERGY.          439 


The  magnetic  energy  density  is  ^H*.  The  velocity  of  the 
magnetic  tubes  at  the  distance  r  (greater  than  a)  from  the  axis  is 

u  =  E\B=  2'jr^rW  00 

if  we  assume  that  (8),  XV.,  and  (6),  XIII.,  apply  to  the  field  of 
the  steady  conduction  current. 

Hence  the  rate  at  which  magnetic  energy  crosses  the  surface 
inward,  or  the  rate  at  which  magnetic  energy  is  dissipated  in  heat 
in  the  conductor,  is,  according  to  the  conception  of  moving  tubes, 

ifjiH2  x  27rrx  u=\WI<1 

Thus  the  energy  dissipated  by  resistance  is,  according  to  this 
conception,  half  electric  and  half  magnetic.  The  same  thing 
may  be  shown  to  be  true  in  the  system  considered  in  §  I,  the 
velocities  of  the  electric  and  magnetic  tubes  being  there,  as  here, 
inversely  proportional  to  the  corresponding  energy  densities 


Within  the  cylinder  the  electric  intensity  is  constant,  the  num- 
ber of  tubes  entering  per  second  being  equal  to  the  rate  at  which 
tubes  are  broken  up.  The  magnetic  tubes  contract  as  they  ap- 
proach the  axis,  thus  giving  up  their  energy  without  being  broken 
up  ;  and  as  the  magnetic  intensity  decreases  the  velocity  of  the 
tubes  increases  in  such  a  way  as  to  make  the  number  of  unit  tubes 
cutting  unit  length  of  a  line  of  electric  intensity  per  second  con- 
stant (E  =  Bu  =  constant).  This  appears  also  from  the  relation 

u  =  EjB  =  27ra2  Wj  prL  (e) 


Inside  the  conductor  (c)  is  of  course  unmeaning. 
On  multiplying  together  (c)  and  (</),  we  see  that 

uv  =  i  fcfJL 
as  in  §  i. 

4,  The  Cartesian  Expressions  for  the  Rectangular  Components 
of  the  Curl  of  a  Vector.  Curl  H  and  Curl  E.  The  Xt  Y,  Z  com- 
ponents of  a  vector  H  being  denoted  by  Hv  H^  Hy  the  X,  Y,  Z 


440    ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

components  of  its  curl  will  be  denoted  by  curl^/7,  cur!2/7,  cur!3/7, 
respectively. 

To  find  the  Cartesian  expression  for  curlx//at  a  point  P(xyy,  z), 
or  i,  Fig.  8,  at  which  the  components  of  //are  Hv  //2,  and  //3, 
we  may  take  the  line  integral  of  H  around  the  infinitesimal  rect- 
angle 1 7,  the  plane  of  which  is  parallel  to  the  KZ  plane  and  the 
sides  of  which,  of  lengths  dz  and  dy,  are  parallel  to  Z  and  Y, 
divide  the  result  by  the  area  of  the  rectangle,  dS^  =  dydz,  and 
pass  to  the  limit. 

This  integration,  which  must  be  performed  in  the  direction  of 
the  arrows  around  the  circuit,  gives 

««!,  =  (Ht  +  \dHJdy  dy}dy  +  [Ht  +  dHJdy  dy 

?sjdy  dy)jdz  dz~\  dz  —  [//2  -f  dH2jdz  dz 

i)jdy  dy\  dy  —  (//3  -j-  \dH^dz  dz)dz 

the  separate  terms  being  the  integrals  along  the  sides  I,  2,  3,  4 
in  the  order  given.  Cancelling  equal  and  opposite  terms,  divid- 
ing by  dSv  and  passing  to  the  limit,  we  obtain 

dClljdSl  =  cur\H  =  dHJdy  -  dH^dz         (a) 


By  an  exactly  analogous  process,  or  by  the  principle 
of  symmetry  and  inspection  of  (a),  we  find 


cur!2//=  dH^dz  —  dH^dx  (ft) 

and  cur!3//=  dHJdx  -  dHJdy  (r) 

From  these  equations  we  may  write  down  at  once  the  com- 
ponents of  the  curl  of  any  other  vector  E.     Thus 

curL/i  =  dEJdy  —  dEJdz  (a)  "1 

1  61       •/  LI  \      / 

cur\2E=dElldz-dEsldx  (b)   L       (8) 

curl^E  =  dE^dx —  d  Eljdy  (c)  J 

5,  The  Flux  of  Electromagnetic  Energy.     Poynting's  Theorem. 
Let  R  denote,  in  both  magnitude  and  direction,  the  time  rate  per 


THE    FLUX   OF   ELECTROMAGNETIC    ENERGY.          441 

unit  area  at  which  electromagnetic  energy  is  transferred  at  a 
point  P  (x,  yy  z)  across  a  surface  normal  to  the  direction  of  trans- 
fer. Let  the  direction  cosines  of  R,  E,  and  H  at  the  point  P  be 
denoted  by  /,  mt  n,  V  ',  mr  ,  nf  ,  I",  m",  n",  respectively  ;  and  let 
the  angle  between  E  and  H  be  denoted  by  6.  E  and  H  will  be 
used  to  denote  the  non-intrinsic  intensities  of  the  field,  intrinsic 
intensities,  when  present,  being  denoted  by  e  and  h.  With  these 
conventions  we  have  (Poynting's  theorem) 

R  =  V£/7sin<9  (9) 

to  the  demonstration  of  which  we  proceed. 

Consider  first  a  region  containing  no  intrinsic  electric  or  mag- 
netic intensity.  From  the  definition  of  R  and  that  of  the  conver- 
gence of  a  vector  it  is  evident  that  the  rate  at  which  electromag- 
netic energy  enters  an  infinitesimal  volume  dr  at  P  through  its 
surface  (minus  the  rate  at  which  energy  leaves  the  volume)  is 
conv  R  •  dr.  Hence,  since  no  electromagnetic  energy  is  de- 
veloped within  dr  (e  =  h  =  o),  this  quantity  is  equal  to  the  rate 
of  increase  of  electromagnetic  energy  plus  the  rate  of  dissipation 
of  energy  in  heat  within  the  volume.  That  is 


conv  R  >dr  =  dr  •  d(\cE*  -f  &&*)/<**  +  dr  -  kE2 


or 


conv  R  =  d(±cE2  +  ^H2)jdt  +  kE2  = 

+  E2dE2jdt  -f  EJEJdt)  +  ^(H^HJdt  (a) 

+  HJHJdt  +  HJHJdi)  -f  J5&  +  £/C2  +  £/C8 


Since 

—  pdHJdt  =  curl^,  etc. 
and 

cdE.  Idt  -f  L  =  curl,//",  etc.  (since  i   —i  =  o) 

I/  Cl  1'  \  CUtti/ 

-    ^^v 

(|l)  may  be  written 

conv  R  =  EL  curlj.tf  +  £,  cur\2ff  +  £3  c\ir\3ff  —  H^  curl^ 

(6) 


442          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

But  also,  by  definition  of  the  convergence  of  a  vector, 
conv  R  =  -  (dRJdx+dRJdy  +  dR^dz) 

Hence      R1  -  #,  =  E2H3  -  EBH2  =  EH(m'n"  -  n'm") 
R2-a2  =  E^  -  E&  =  EH(n'l"  -  I'n"} 
R3  -  as  =  E^H2  -  E^  =  EH(l'm"  -  m'l") 

where  a,  with  components  av  av  ay  is  a  vector  whose  conver- 
gence is  zero. 

Without  affecting  in  any  way  the  generality  of  the  conclusions, 
we  may,  for  simplicity,  give  the  rectangular  coordinate  system 
such  an  orientation  as  to  make  the  plane  XY  parallel  with  the 
plane  containing  E  and  Ht  and  the  direction  of  X  coincident  with 
that  of  E.  Then  El  =  E,  E2  =  E3  =  o,  or  I'  =  /,  m'  =  n'  =  o ; 
and  7/3  =  o,  or  n"  =  o,  while  I"  =  cos  0  and  m"  =  cos  (90°  —  6) 
—  sin  6.  With  these  simplifications  the  above  equations  become 

R!  —  al  =  o  =  R2  —  av  and  R3  —  a3  =  R  —  <?  =  £77  sin  6 
Hence,  with  due  respect  to  the  directions  of  the  vectors, 

R  =  MEH  sin<9  +  tf  (10) 

Since  div  a  =  conv  a  =  o,  a,  if  it  is  not  zero,  represents  a  flux 
of  energy  in  closed  tubes  and  therefore  contributes  nothing  to  the 
net  energy  entering  any  volume.  In  what  follows  this  circuitous 
energy  flux  will  be  neglected,  or,  what  amounts  to  the  same 
thing,  a  will  be  assumed  equal  to  zero,  unless  the  contrary  is 
stated.  With  this  assumption,  (10)  is  identical  with  (9). 

If  at  any  point  P  there  is  an  intrinsic  electric  intensity  e  and 
an  intrinsic  magnetic  intensity  h  in  addition  to  the  field  intensities 
E  and  //,  then  an  element  of  volume  at  Pt  in  addition  to  receiv- 
ing electromagnetic  energy  by  transference  across  its  surface  at 
the  time  rate  conv  R  per  unit  volume  or  R  per  unit  area,  receives 
electromagnetic  energy  by  transformation  on  the  spot  at  the 
time  rate 

ei  cos  B1  +  kdBjdt-cos  6" 


THE    FLUX   OF   ELECTROMAGNETIC    ENERGY.          443 

per  unit  volume,  where  9f  denotes  the  angle  between  e  and  it 
and  6"  that  between  h  and  dB\dt. 

The  vector  R  is  called,  as  stated  in  §  I,  the  electromagnetic 
energy  flux  density. 

Since  R  is  perpendicular  to  E  and  to  H,  the  lines  along  which 
the  energy  flows,  or  the  energy  stream-lines,  are  the  intersections 
of  the  electric  and  magnetic  equipotential  surfaces.  (Even  when 
the  field  is  not  static  or  steady,  so  that  the  term  potential  cannot 
be  used  legitimately,  we  may  still  use  the  expression  equipotential 
surface  to  denote  a  surface  perpendicular  at  every  point  to  the 
intensity.) 

6.  A  Long  Circular  Cylindrical  Conductor  Traversed  by  a  Steady 
Current.  Fig.  134  shows  a  section  of  a  small  jpart  of  the  electric 
field  within  and  without  the  conductor  when  the  current  has  the 


Fig.  134. 

direction  AB.  The  lines  of  magnetic  intensity  are  circles  about 
the  axis  AB  of  the  cylinder,  going  down  into  the  paper  below 
AB  and  coming  up  out  of  the  paper  above  AB.  The  electric 
and  magnetic  intensities  are  everywhere  perpendicular. 

Outside  the  wire,  the  energy  flux  density  R  has  the  direction 
indicated  at  C,  with  a  component  Rt  in  the  direction  of  the  axis 
AB,  and  a  component  R2  toward  the  axis.  Thus  in  the  dielectric 
energy  is  transferred  in  the  direction  AB  to  parts  of  the  field  far- 
ther along  the  circuit  by  the  component  Ry  and  energy  is  also 


444          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

transferred  toward  and  into  the  conductor  by  the  component  R2. 
Within  the  conductor,  as  at  D,  R  is  directed  toward  the  axis 
with  no  component  parallel  to  the  axis.  Hence  within  the  con- 
ductor there  is  no  transfer  of  energy  along  the  circuit,  all  the 
energy  entering  the  conductor  from  the  dielectric  and  being  dis- 
sipated in  heat  by  resistance. 

If  a  conductor,  either  insulated  from  the  circuit  or  connected 
thereto  at  a  single  point,  is  placed  in  the  electromagnetic  field, 
then  there  will  be  a  magnetic  field  inside  the  conductor,  but  no 
electric  field,  and  the  tubes  of  electric  displacement  will  terminate 
normally  upon  the  outer  surface  of  the  conductor.  Within  the 
conductor  there  is  no  transference  of  energy,  since  E  is  zero.  In 
the  dielectric  at  the  surface  of  the  conductor  the  energy  stream- 
lines are  parallel  to  its  surface,  since  they  are  perpendicular 
to  E,  which  is  normal  to  the  surface.  Thus  the  energy  streams 
around  the  conductor  as  a  liquid  streams  around  an  impervious 
solid. 

7,  The  Transfer  of  Energy  in  and  About  a  Voltaic  Cell  and  a 
Simple  Electrolytic  Cell.  Figs.  135-138  represent  diagrammatic- 
ally  for  several  cases  the  electric  field  and  the  transfer  of  electro- 
magnetic energy  in  and  about  a  Daniell  cell  under  the  assump- 
tions (for  which  evidence,  though  not  wholly  satisfactory,  can  be 
adduced)  that  the  single  difference  of  potential  from  the  copper 
electrode  to  the  copper  sulphate  solution  is  positive  and  equal 
to  that  from  the  zinc  sulphate  solution  to  the  zinc,  and  that  the 
single  difference  of  potential  from  the  copper  sulphate  solution 
to  the  zinc  sulphate  solution  is  negligible.  ABC  represents  the 
copper  electrode,  HIJ  the  zinc  electrode,  DEFG  the  solutions, 
and  the  dotted  line  the  porous  partition  between  them.  The 
distance  between  the  electrolyte  and  the  electrodes  is  of  course 
enormously  exaggerated  in  the  diagrams.  The  intrinsic  electro- 
motive forces  are  directed  from  the  copper  sulphate  solution  to 
the  copper  and  from  the  zinc  to  the  zinc  sulphate  solution  exactly 
opposite  to  the  electric  fields  they  develop. 


THE   FLUX   OF    ELECTROMAGNETIC    ENERGY. 


445 


The  electric  field  on  open  circuit  is  shown  in  Fig.  135.  There 
is  no  electric  intensity  within  the  conductors,  no  current,  no  mag- 
netic field,  and  no  transfer  of  energy. 

The  field  after  closing  the  circuit  above  A  and  H  is  shown  in 
Fig.  136.  Above  BI  the  magnetic  intensity  is  directed  (in  the 
plane  of  the  diagram)  normally  into  the  paper,  while  below  BI  it 
is  directed  up  out  of  the  paper.  The  'direction  of  the  transfer- 
ence of  electromagnetic  energy  is  shown  by  the  arrows  cutting 


Fig.  135. 


Fig.  136. 


the  lines  of  intensity  normally.  Electromagnetic  energy,  trans- 
formed from  chemical  energy  with  the  deposition  of  copper  and 
the  solution  of  zinc  at  the  electrodes,  moves  out  from  between  the 
electrolyte  and  the  electrodes  into  the  dielectric,  part  there  con- 
verging upon  and  moving  into  the  electrolytic  and  metallic  con- 
ductors, there  to  be  dissipated  in  Joulean  heat,  and  part  being 
carried  into  other  parts  of  the  field. 

The  field  and  transference  of  energy  when-an  agent  with  a  con- 
siderably greater  e.m.f.  than  that  of  the  given  cell  sends  a  cur- 
rent through  it  (or  assists  in  so  doing)  in  the  same  direction  as 
befrre  is  shown  in  Fig.  137.  Electrical  energy  generated  by  the 


44^ 


ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


external  e.m.f.,  as  well  as  all  the  energy  generated  by  the  in- 
trinsic e.m.f.  of  the  cell  itself,  is  dissipated  in  the  conductors. 

The  field  and  the  transference  of  energy  when  a  current  is  sent 
through  the  cell  in  opposition  to  its  intrinsic  e.m.f.,  that  is  from 
copper  to  zinc,  is  represented  in  Fig.  138.  Here  a  portion  of 
the  energy  supplied  to  the  cell  by  the  external  e.m.f.  is  dissipated 
in  the  conductors  and  the  rest  is  transformed  into  chemical  en- 
ergy with  the  deposition  of  zinc  at  the  kathode  and  the  solution 
of  copper  at  the  anode. 


Fig.  137. 


Fig.  138. 


In  Fig.  139  the  electric  field  and  the  transference  of  energy  in 
and'  about  an  electrolytic  cell,  consisting  of  copper  electrodes 
AC  and  ///dipping  in  a  solution  DEFG  of  copper  sulphate,  are 
represented  diagrammatically  for  the  case  in  which  a  current 
traverses  the  system  from  A  to  H.  Between  the  kathode  HJ 
and  the  electrolyte  energy  is  transformed  from  chemical  to 
electrical  with  deposition  of  copper,  whence  it  moves  out  into  the 
dielectric,  and  thence  partly  into  the  conductors  and  partly  into 
the  region  between  the  anode  A  C  and  the  electrolyte,  where  re- 
conversion into  chemical  energy  occurs.  The  quantity  of  chemi- 
cal energy  transformed  into  electrical  at  the  kathode  is  equal  to 
the  quantity  of  electrical  energy  transformed  into  chemical  energy 


THE    FLUX   OF    ELECTROMAGNETIC    ENERGY. 


447 


at  the  anode.  Hence,  since  a  portion  of  the  electrical  energy 
generated  at  the  kathode  is  dissipated  in  heat,  as  much  electrical 
energy  coming  from  the  external  e.m.f.  producing  the  current  is 
transferred  into  the  region  about  the  anode,  and  there  converted 


Fig.  139. 

into  chemical  energy,  as  is  dissipated  in  the  conductors  of  the 
electrical  energy  generated  at  the  kathode.  On  the  whole,  no 
work  is  done  in  the  cell  except  that  done  upon  resistance.  (If 
the  electrodes  are  at  different  levels,  work  will  be  done  by  or 
against  gravity  during  the  conduction,  etc.) 

8.  A  Circuit  Containing  a    Motional    E.M.F.     We  shall  con- 
sider only  the  simple  case  of  a  slider,  AB,  Fig.  140,  running  on 


BU 


Fig.  140. 


two  parallel  rails  A  C  and  BDy  connected  by  a  cross  piece  CD  and 
immersed  in  a  uniform  magnetic  field  directed  downward  into  the 


448          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

plane  of  the  paper.  Let  AB  move  to  the  right.  Then  there  is 
induced  in  AB  an  intrinsic  e.m.f.  in  the  direction  BA,  producing 
a  current  around  the  circuit  in  the  direction  BACDB,  with  lines 
of  magnetic  intensity  related  to  the  current  in  the  usual  manner. 
The  intrinsic  e.m.f.  in  the  direction  BA  produces  an  electric  field 
intensity  with  the  general  direction  from  A  to  B  along  the  slider 
and  around  the  circuit,  both  within  and  without  the  conductor. 
The  lines  of  electric  intensity  diverge  from  the  upper  half  of  AB 
in  the  dielectric  and  converge  upon  the  lower  half.  Hence,  since 
R  =  VEH  sin  6,  the  electromagnetic  energy  generated  in  AB 
moves  outwards  from  AB  and  toward  its  ends,  then  contracts 
upon  the  rest  of  the  circuit,  all  the  energy  being  finally  dissipated 
in  heat.  While  the  energy  is  moving  outward  through  the  con- 
ductor AB  a  part  of  the  energy  is  dissipated  owing  to  the  resist- 
ance of  AB,  not  all  the  energy  developed  in  AB  reaching  the 
dielectric. 

9.  A  Concentrated  Electric  Charge  in  the  Presence  of  a  Concen- 
trated Magnetic  Pole.     If  the  charge  and  pole  are  concentrated 
at  two  points  A  and  B,  respectively,  and  if  the  energy  flux  den- 
sity is  given  by  MEH  sin  6,  the  energy  stream-lines  are  circles 
about  AB  and  AB  produced   as  axis.      Since  both   fields  are 
purely  static  in  this  case,  however,  there  is  no  reason  to  believe 
that  any  flow  of  energy,  even  in  closed  tubes,  exists.     To  recon- 
cile this  view  with  Poynting's  theorem,  we  have  only  to  remem- 
ber that  the  energy  flux  density,  in  the  general  case,  is  VEH 
sin  0  plus  a  circuital  flux  density  a,  and  to  suppose  that  in  the 
present  case  a  =  —  V 'EH  sin  0,  or  R  =  o. 

10.  Electric  Radiation.     Electric  Waves.     The  damping  of  the 
mechanical  vibrations  described  in   §  45,  III.,  C,  XIII.,  was  as- 
sumed to  be    due  wholly   to  friction.     A   vibrating  mechanical 
system,   however,  unless    completely  surrounded    by  a   perfect 
vacuum,  which  is  not  possible,  will  set  the  adjacent  parts  of  the 
surrounding  medium  into  vibration,  thus  emitting  a  train  of  waves 


THE    FLUX    OF    ELECTROMAGNETIC    ENERGY.          449 

Owing  to  the  energy  thus  radiated  to  the  surrounding  medium, 
the  motion  of  the  system  will  be  damped,  and  the  damping  so 
caused  may  greatly  exceed  the  damping  due  to  friction.  This  is 
true,  for  example,  in  the  case  of  a  vibrating  air  column,  most 
of  whose  energy  is  emitted  in  waves  of  sound. 

O*' 

Other  things  being  equal,  it  is  clear  that  the  damping  due  to 
radiation  will  be  greater  the  greater  the  surface  communicating 
the  energy  to  the  surrounding  medium. 

The  damping  of  the  electrical  oscillations  discussed  in  §  43  C, 
XIII.,  was  also  assumed  to  be  due  wholly  to  the  dissipation  of 
energy  by  resistance.  But  since  the  electromagnetic  field  of  the 
system  extends  into  all  space,  it  is  evident  that  when  its  oscil- 
lations, or  variations  in  the  nearer  portions  of  its  electromagnetic 
field,  occur,  a  train  of  electromagnetic  waves  must  be  emitted  by 
the  system  and  propagated  into  space,  and  that  the  oscillations 
will  therefore  be  damped  owing  to  the  energy  thus  radiated. 


r— 

E 
To  Induction 

—1 

Coil 
Fig.  141. 

Other  things  being  equal,  it  is  clear  that  the  damping  due  to 
radiation  will  be  greater  the  farther  into  space  the  stronger  parts 
of  the  system's  electric  and  magnetic  fields  extend.  Thus  the 
system  shown  in  Fig.  141,  in  which  the  fields  spread  far  out  into 
the  surrounding  dielectric,  is  a  good  radiator,  the  damping  due  to 
radiation  being  so  great  that  only  a  few  oscillations  are  com- 
pleted ;  while  a  system  such  as  that  of  Fig.  142,  whose  electric 
field  is  confined  almost  wholly  to  the  region  AB,  and  whose 
magnetic  field  is  confined  almost  wholly  to  the  region  CD,  is  a 
good  vibrator,  but  a  poor  radiator,  its  energy  being  completely 
radiated  into  space  and  dissipated  in  its  resistance  only  after  the 
execution  of  many  oscillations. 


450          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 


At  a  considerable  distance  from  an  electrical  oscillator  the 
electromagnetic  waves  crossing  a  limited  area  will  be  approxi- 
mately plane,  just  like  the  sound  waves  emitted  by  a  vibrating 
bell  or  diapason* 

In  the  electromagnetic  wave  train  emitted  by  a  symmetrical 
"  dumb-bell "  oscillator  like  that  of  Fig.  141,  consisting  of 
spheres  at  the  ends  of  circular  cylindrical  rods,  it  is  obvious  that 
at  any  point  the  electric  intensity  will  lie  in  the  plane  containing 


A 


\ 


E 

To  Induction 
Coil 

Fig.  142. 


the  point  and  passing  through  the  axis  of  the  oscillator,  and 
the  magnetic  intensity  will  lie  in  the  plane  perpendicular  to  the 
axis,  both  intensities  being  perpendicular  to  the  direction  of  propa- 
gation of  the  waves  by  Poynting's  theorem. 

Such  a  wave  train  passing  through  a  given  point  at  which  the 
electric  and  magnetic  intensities  oscillate  in  fixed  planes  is  said 
to  be  plane  polarised. 

Electromagnetic  waves  can  also  be  developed  by  means  of  the 
convection  of  electric  charges  or  magnetic  poles.  Thus  if  two 
eqwa^r  spheres,  with  equal  and  opposite  charges,  are  made  to 
approach-and  recede  from  one  another  alternately,  a  wave  system 
very  similar  to  that  of  the  "  dumb-bell  "  oscillator  will  be  emitted. 

Also,  if  a  system  consisting  of  two  equal  spheres  with  equal 
and  opposite  charges,  mounted  upon  an  insulating  rod,  is  rotated 
uniformly  about  an  axis  passing  perpendicularly  through  the 


THE   FLUX   OF    ELECTROMAGNETIC    ENERGY.          451 

center  of  the  rod,  another  interesting  and  important  wave  system 
will  be  emitted.  At  any  point  along  the  axis  of  revolution  each 
intensity,  always  perpendicular  to  this  axis,  remains  constant  in 
magnitude,  but  passes  uniformly  through  all  azimuths  during 
every  revolution.  The  radiation  along  the  axis  is  therefore  said 
to  be  circularly  polarised.  At  all  points  in  the  plane  of  revolu- 
tion of  the  centers  of  the  spheres,  the  radiation  is  plane  polarised. 
At  points  not  in  this  plane  and  not  on  the  axis  of  revolution, 
each  intensity  passes  uniformly  through  all  azimuths  during  each 
period,  reaching  a  minimum  and  a  maximum  value  twice  each, 
but  never  becoming  zero,  and  the  radiation  is  therefore  said  to  be 
elliptically  polarised.  The  first  and  second  cases  are  particular 
cases  of  the  third. 

On  the  electron  theory,  waves  of  light,  which  are  electromag- 
netic waves  of  extremely  short  wave-length  and  period,  are  devel- 
oped by  the  vibrations  of  the  electrons,  that  is  by  electric  con- 
vection, within  the  atom. 

For  the  continuous  production  of  electric  waves,  or  rather  for 
the  rapid  production  of  successive  trains  of  such  waves,  the  two 
conductors  of  the  oscillator  are  connected  permanently  to  the 
terminals  of  an  electrical  influence  machine  or  induction  coil  in 
operation,  as  shown  in  the  figures.  Every  time  the  voltage  be- 
tween the  terminals  of  the  oscillator  reaches  a  certain  value,  the 
insulating  properties  of  the  dielectric  break  down  along  a  line 
between  the  terminals,  and  the  oscillations  occur,  the  path  of  the 
current  being  evident  from  the  spark.  With  the  cessation  of  the 
oscillation  the  insulation  is  restored,  the  voltage  again  increases,  a 
spark  occurs,  another  wave  train  is  emitted,  and  so  on  indefinitely. 

For  the  detection  of  electric  waves,  any  sufficiently  sensitive 
electric  vibrator  may  be  employed.  When  used  for  this  purpose, 
such  an  electrical  system  is  called  a  resonator.  One  of  the  com- 
monest forms  of  resonator  is  the  dumb-bell  form,  similar  to  the 
dumb-bell  vibrator,  Fig.  141,  but  with  a  shorter  spark  gap  E. 
If  such  a  resonator  is  placed  in  a  region  traversed  by  electric 
waves  with  the  rods  EF  parallel  to  the  direction  of  the  electric 


45  2          ELEMENTS   OF    ELECTROMAGNETIC   THEORY. 

intensity,  or  in  any  direction  not  perpendicular  to  the  intensity, 
an  e.m.f.  of  the  same  kind  as  that  of  the  vibrator  will  be  im- 
pressed upon  the  resonator  parallel  to  its  length,  and  oscillations 
will  be  set  up  therein.  If  the  maximum  value  of  the  voltage  de- 
veloped between  the  terminals  of  the  spark  gap  E  is  sufficient, 
the  insulator  within  the  gap  will  break  down  and  the  oscillations 
of  the  resonator  will  become  manifest  by  the  passage  of  a  spark. 
This  effect  will  be  a  maximum  when  the  rods  EF  are  parallel  to 
the  electric  intensity  of  the  waves,  and  zero  when  they  are  per- 
pendicular to  this  intensity.  For  a  given  angle  between  the 
electric  intensity  and  the  axis  of  a  resonator,  the  effect  will  also 
be  a  maximum  when  the  period  of  the  resonator  is  equal  to  that 
of  the  waves  (or  that  of  the  vibrator),  in  accordance  with  the 
principles  of  §  44,  XIII.  One  tuning  fork  set  into  resonant  vi- 
bration by  the  waves  from  another  in  unison  is  an  exact  mechan- 
ical analogue. 

The  sensitiveness  of  a  resonator  can  be  greatly  increased,  or 
the  minimum  intensity  for  which  it  will  give  noticeable  indications 
greatly  diminished,  by  the  addition  of  any  one  of  several  devices. 

One  of  the  most  effective  and  widely  used  of  these  adjuncts 
is  the  coherer  of  Branly.  This  consists  of  a  small  glass  tube 
plugged  at  the  ends  with  metallic  electrodes  and  loosely  packed 
with  metallic  filings,  or  other  small  pieces  of  metal.  The  elec- 
trodes are  connected  to  the  resonator,  usually  on  opposite  sides 
of  the  spark  gap,  and  also  to  the  terminals  of  a  circuit  contain- 
ing a  battery  and  a  galvanometer  or  other  current  indicator. 
Before  the  incidence  of  electric  waves  upon  the  resonator,  the 
electric  resistance  of  the  coherer  is  very  great  and  only  a  very 
small  current  traverses  the  galvanometer.  But  when  oscillations 
are  set  up  in  the  resonator  by  the  impact  of  electric  waves,  the 
resistance  of  the  coherer  is  greatly  diminished,  and  the  current 
through  the  galvanometer  is  therefore  greatly  increased.  The 
resistance  of  the  coherer  retains  its  reduced  value  after  the  cessa- 
tion of  the  waves,  but  the  original  high  resistance  can  be  imme- 
diately restored  by  tapping  the  instrument  with  a  light  hammer. 


THE   FLUX   OF   ELECTROMAGNETIC    ENERGY.         453 

In  practice  this  is  done  by  a  continuously  operating  hammer 
driven  electrically  by  a  separate  circuit.  The  diminution  of  re- 
sistance on  which  the  action  of  the  coherer  depends  is  probably 
brought  about  by  the  welding  together  of  the  metallic  particles 
on  the  passing  of  very  minute  sparks  between  them  when  even 
very  feeble  oscillations  are  set  up  in  the  resonator.  When  the 
particles  are  shaken  apart  by  the  hammer,  the  resistance  goes 
back  to  its  previous  magnitude. 

An  indication  of  the  relative  magnitude  of  the  electric  intensity 
is  given  by  the  maximum  distance  between  the  spark  gap  termi- 
nals of  the  resonator  (which  are  made  adjustable  when  the  in- 
strument is  used  for  this  purpose)  at  which  sparking  will  occur ; 
or,  if  a  coherer  is  employed,  by  the  diminution  of  resistance,  or 
increase  of  galvanometer  current,  produced. 

For  a  detailed  account  of  the  theory  of  electric  waves  and 
oscillations  and  of  the  extensive  experimental  investigations  (in 
full  accord  with  the  theory)  upon  the  subject,  the  reader  is  re- 
ferred to  Poincare's  Les  Oscillations  Electriques,  J.  J.  Thomson's 
Recent  Researches  in  Electricity  and  Magnetism,  Winckelmann's 
Handbuch,  and,  for  recent  digests,  to  the  Rapports  of  the  Inter- 
national Congress  of  Physics,  Vol.  II.  The  most  complete  treat- 
ment of  the  theory  of  the  propagation  of  waves  along  wires  is  con- 
tained in  Heaviside's  Electromagnetic  Theory  and  Electrical  Papers. 
For  the  electromagnetic  theory  of  light,  see  Drude's  Optik. 

The  following  paragraphs  contain  the  theory  of  some  of  the 
simplest  and  most  fundamental  phenomena  of  electric  waves. 

11.  The  Propagation  of  Electromagnetic  Disturbances  in  a  Non- 
Conducting  Dielectric  Containing  no  Other  Electric  or  Magnetic 
Fields  than  Those  of  the  Disturbance  Itself.  In  this  case  the 
electric  convection  and  conduction  current  densities  are  zero,  and 
e  and  /n  are  constant  in  space  and  time.  Hence  the  first  and 
second  laws  of  circuitation  are 

i  =  dD\dt  =  cdEjdt  =  curl  H  ( 1 1 ) 

and 

-  dB\dt  =  iJLdHjdt  =  curl  E  (12) 


454          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

(i  i)  is  equivalent  to  the  three  component  equations 

cdEJdt  =  dHjdy  -  dHJdz  (a) 

cdEJdt  =  dHJdz  -  dHJdx  (b) 

cdEJdt  =  dHJdx  -  dHJdy  (c) 

and  (12)  is  equivalent  to  the  three  equations 

=  dEJdy  -  dEJdz  (^ 

=  dE^  jdz  -  dEJdx  (b) 

(c) 


(13) 


(  1  4) 


Simple  Plane  Wave.  We  shall  consider  first  only  the  simplest 
of  plane  polarised  electromagnetic  waves,  viz.,  a  disturbance  in 
which  everywhere  E2  =  Ez  =  o,  and  E^=  E)  is  independent  of 
x  and  y,  or  has  the  same  magnitude  and  direction  at  all  points 
of  any  plane  distant  z  from  XY  plane,  i.  e.,  a  plane  polarised 
plane  wave.  In  this  case  (13)  (a)  becomes 


dDJdt(=  dDj  df)  =  cdEJdt(=  cdEfdt)  =  -  dHJdz     (15) 

since  the  magnetic  intensity  must  be  independent  of  x  and  y  when 
the  electric  intensity  is  independent  of  x  and  y  ;  and  (14)  (b)  be- 
comes 

dB2ldt  =  ^dHjdt=  -dEJdz  (16) 

Differentiating  (15)  with  respect  to  /  and  (16)  with  respect  to 
z,  and  combining  the  resulting  equations,  we  obtain 

(17) 


tPDJdt*  =i/w  d*DJd£  (18) 

Differentiating  (15)  with  respect  to  z  and  (16)  with  respect  to 
/  and  combining  the  resulting  equations,  we  obtain 


(  1  9) 
d2B2jdt2  =  i  jcii,  •  d*BJdz2  (20) 

These  four  equations  have  all  the  same  form  and  show  that 
the  electric  and  magnetic  intensities  and  inductions  are  propa- 


THE   FLUX    OF    ELECTROMAGNETIC    ENERGY.          455 

gated  in  a  direction  parallel  to  the  axis  of  Z  with  the  velocity 

v=il(citf  (21) 

To  demonstrate  this,  we  have  only  to  obtain  the  general  solu- 
tion of  one  of  the  equations.  Choosing  (17),  introducing  two 

new  variables 

a  =  z  —  vt 
and 

b  =  z  -f  vt 

where  v  is  given  by  (21),  and  eliminating  z  and  /  from  (17)  by 
means  of  these  equations,  we  have 

d2EJdadb  =  o,  or  djda  (dEJdb)  =  djdb  (dEJda)  =  o 

Hence  dE^da  is  a  function  of  a  only,  and  dE^db  is  a  function 
of  b  only.  Therefore  E{  consists  of  the  sum  of  two  terms,  one 
a  function  of  a  only,  and  the  other  a  function  of  b  only.  Thus 
the  general  solution  of  (17)  is 

El  =  Ffc  -  vt}  +  F&s  +  vt)  (2  2) 

where  F^z  —  vf)  and  F2(z  +  vt)  are  arbitrary  functions  of  (z  —  vt) 
and  (z  +  ztf),  respectively.  Either  function  may  be  zero,  but 
neither  can  be  constant  or  contain  a  constant  term,  since  a  con- 
stant field  is  excluded  by  the  conditions  assumed  above. 

FJz  —  vt)  represents  a  disturbance  in  the  dielectric  traveling 
unchanged  in  form  in  the  positive  direction  of  Z  with  the  velocity 
v.  For  at  the  time  /  -f  /',  Fl  has  the  same  value 

Fl  [(*  -f  vf)  -  v(t  +  /')]  =  FI(*  ~  vt) 

at  the  plane  whose  Z  coordinate  is  z  -\-vt'  which,  at  the  time  /*, 
it  had  at  the  plane  whose  Z  coordinate  is  z. 

Similarly,  F2(z  -f  vt)  represents  a  disturbance  traveling  in  the 
negative  direction  of  Z  with  the  same  speed  v. 

At  the  time  /  =  o,  (22)  gives 

:          '  £0  =  F^  +  F^)  (23) 

If  at  the  time  t—ot  dEJdt=  o,  or  the  initial  electric  field  is 
static,  (22)  gives  also 

^.'W-^.'W  (24) 


456 


ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 


Hence  we  have  at  the  time  t  =  o  in  this  case,  by  integrating 
(24)  and  making  use  of  (23), 

-  .  F&i)  =  FJd-\E,    -*«c^>e    (25) 

there  being  no  constant  of  integration,  since  there  is  no  perma- 
nent field. 

As  an  example,  suppose  that  at  the  time  t=oyEl=Dljc  — 
2 A  cos  27TZJL  between  the  limits  z  =  +  Z/4  and  z  =  —  £/4, 
and  El  =  o  everywhere  else  ;  also  that  at  the  same  time  dE^dt 
=  o  everywhere.  Then 

F^z)  =  A  cos  2irzjL  =  F&s)  =  F(z) 
and  at  any  time  / 

El  =  E=  F(s  -  vf)  +  F(z  +  vt) 

=  A   COS    27T/L  •(%  —  Vt)  +  A   COS   27T/L  •  (Z  -f  Vt) 

Thus  at  the  time  /  =  o  the  initial  static  displacement  or  in- 
tensity divides  up  into  two  equal  waves,  one  running  in  the  posi- 
tive direction  of  Z  with  the  speed  v,  and  the  other  running  in  the 


negative  direction  of  Z  with  the  same  speed.  At  the  time  /  the 
intensity  is  zero  everywhere  except  between  the  planes  #  =  vt 
-f  L/4  and  z  =  vt  —  L/4  and  between  the  planes  z  =  —  (vt  + 
L/4)  and  z  —  —  (vt  —  LJ4).  The  initial  disturbance  and  the 
disturbance  at  the  time  /  are  shown  in  Fig.  143. 


THE   FLUX   OF    ELECTROMAGNETIC    ENERGY.          457 

12,  The  Relation  Between  E  and  H  in  the  Electromagnetic 
Wave.  From  the  exact  similarity  in  the  form  of  equations  (17) 
.and  (19)  it  is  now  evident  that 

+*<)  (26) 


where  fL  and  f.2  are  arbitrary  functions  of  (z  —  vf)  and  (z  -f  vt\ 
respectively. 

H2  is  the  total  magnetic  intensity.      For  since   £2  =  Es  =  o, 
and  E^  =  E  is  independent  of  x  and  y,  (  1  4),  (a)  and  (c)  become 


which  gives 

/f.-./r.-o 

the  constant  of  integration  being  zero,  since  there  is  no  perma- 
nent field. 

The  arbitrary  functions  /x  and  /2  are  closely  related  to  Fl  and 
Fv  as  will  appear  from  the  following  deduction  of  H2  —  H  from 
(16)  and  (22).  From  (16)  we  have 


^jdz  dt  (27) 

there  being  no  constant  of  integration.     Consider  the  disturbance 

E^F^-vi)  (28) 

traveling  with  the  velocity  v  in  the  positive  direction  of  Z.     Dif- 
ferentiation of  (28)  gives 


which,  substituted  in  (27),  gives  for  the  disturbance  /j  connected 
with  F^ 


^z  —  vt)jdt  dt  =  i/pv  •  FJ(JS  —  vf) 


In  exactly  the  same  way  we  obtain  for  the  disturbance  f2  con- 
nected with  F9 


458          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 

H2  =f*(2  +  «*)  =  I/K  -  v)  •  F&  +  vt)  =  -  I  IILV  •  F2(z  +  vi)  (30) 

which  is  exactly  analogous  to  (29),  since  the  velocity  of  the  dis- 
turbance f£z  -f  vi)  and  /^(^  -f  vf)  is  —  z>. 

Thus  in  the  general   case,  when    E=El  is  given  by  (22), 


H~H  -/*-*/  +/2(*  +  */) 


+  ztf),  for  the  times  t  =  o  and  ^  =  /,  is  shown 
in  Fig.  143  for  the  case  in  which  F^(z)  =  F2(s)  =  A  cos  27rz/L. 
Since  f^z)  =  —f2(z)  =  cvF^z),  the  initial  magnetic  disturbance 
is  zero. 

With  due  regard  to  the  directions  as  well  as  to  the  magnitudes 
of  the  electric  and  magnetic  intensities  and  the  velocity  of  the 
electromagnetic  disturbance  (Fl  and  flt  or  F2  and  /2),  which  will 
be  denoted  by  v,  both  (29)  and  (30)  give  the  relations 


c\/vEl  =  cVvE  =  MvD  (3  2) 

and 

VBv  (3  3  ) 


since  v*  = 

Quantitatively,  either  of  these  equations  is  equivalent  to  the 
relation 

(34) 


between  the  electric  and  magnetic  energy  densities. 

The  electromagnetic  energy  flux  density  at  any  point  of  the 
wave  is 

R  =  \IEH=  v(±cE2  +  J/^2)  (35) 


and  has  always  the  direction  of  the  velocity  of  the  wave. 

13.  A  Plane  Simple  Harmonic  Wave  Train,     As  another  ex- 
ample, we  shall  assume 

E  =  El  =  A  cos  27T/Z  •  (vt  —  z)  (36) 

Then  we  have 

H=H2  =  cvA  cos  2TrlL'(yt—z)  (37) 


THE   FLUX   OF    ELECTROMAGNETIC    ENERGY.          459 

The  relation  between  E,  Hy  and  the  time,  for  the  plane  z  =  o, 
as  well  as  the  relation  between  E,  H,  and  zt  for  the  time  t  =  o, 
is  shown  in  the  curves  of  Fig.  144. 

The  electromagnetic  energy  flux  density  at  the  time  /  across  a 
plane  distant  z  from  the  XY  plane  is 

(38) 


R  =  V£77=  cvA*  cos2  27T/L  •  (vt  —  z) 
E  and  H  being  given  by  (36)  and  (37). 


,-H 


Z  and  t 


As  shown  by  the  equation,  R  is  always  positive  except  at 
points  at  which  E  and  Ht  and  therefore  MEH=  R,  are  zero.  This 
is  of  course  obvious,  since  the  energy  travels  with  the  waves. 

The  mean  value  of  V  at  any  point  during  a  complete  period  is 

(R)  =  cvAz  x  mean  value  of  cos2  2irjL  •  (vt  —  z) 

=  cvA2  x  mean  value  of  (39) 

1  [l   —  COS  47T/L  •  (Vt  —  Z)\   =  CVA*\2 

Since  in  a  pure  electromagnetic  wave  the  electric  and  magnetic 
intensities  travel  with  the  same  velocity  v,  the  above  result  may 
also  be  obtained  from  the  relation 


R  = 


(40) 


460          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

The  mean  value  of  E2  is  A2  /2,  and  the  mean  value  of  H2  is 
<?v*A2J2.  Hence  (R)  =  cvA2J2,  as  before. 

14.  A  Plane  Electromagnetic  Wave  Train  in  a  Non-conducting 
Dielectric  Incident  and  Reflected  Normally  at  a  Plane  Interface 
Separating  the  Dielectric  from  (1)  a  Perfect  Conductor  or  a  Di- 
electric of  Infinite  Permittivity,  or  (2)  a  Perfect  Insulator  with 
Infinite  Inductivity.  (i)  Reflection  from  a  perfect  conductor.  Let 
the  electric  intensity  in  the  incident  wave  train  be  denoted  by 

Eu  =  A  cos  (nt—pz)  (41) 

the  incident  wave  being  propagated  in  the  positive  direction  of  Z. 
If  EH  were  the  total  intensity,  the  intensity  at  the  interface 
would  be  A  cos  nty  a  quantity  differing  from  zero  except  at  two 
instants  in  every  period.  But  since  the  conductivity  of  the  con- 
ductor, or  the  permittivity  of  the  second  dielectric,  is  infinite,  a 
finite  intensity  parallel  to  the  interface  would  necessitate  an  infi- 
nite current  in  the  conductor,  according  to  Ohm's  law,  or  an 
infinite  displacement  in  the  second  dielectric,  which  is  inadmis- 
sible. Hence  there  must  be  a  reflected  wave  train  whose  in- 


(42) 

added  to  the  intensity  Eu  of  the  incident  train  makes  the  total  in- 
tensity 

El  =  Eli  +  Elr  =  A  [cos  (nt  -  pz)  -  cos  (nt  +  /*)]        (43) 

a  quantity  equal  to  zero,  when  z  =  o,  for  all  values  of  t. 
The  total  magnetic  intensity  is 

H2  =  HK  +  H2r  =  p^n  •  A  [cos  (nt  -  pz)  +  cos  (nt  +  /*)]    (44) 
Equations  (43)  and  (44)  may  be  written 

E^  =  2A  sin  nt  sin  pz  =  2A  sin  27r//7"-sin  2irz  /  L     (45) 
and 

H2  =  2ApjfJLH  •  COS   27Tt/  T-  COS   2TTZJL  (46) 

Thus  the  incident  and  reflected  wave  trains  interfere  to  produce 
a  system  of  standing  waves.     The  electric  nodes,  or  points  (planes) 


THE   FLUX   OF   ELECTROMAGNETIC   ENERGY.         461 

at  which  the  electric  intensity  and  displacement  are  permanently 
zero,  are  distant  from  the  interface  o,  L/2,  L,  $L/2,  etc.;  and  the 
antinodes  y  or  points  (planes)  at  which  the  electric  intensity  reaches 
its  maximum  and  minimum  values,  are  distant  from  the  inter- 
face L/4,  $L/4,  etc. 

The  magnetic  nodes  have  the  positions  of  the  electric  antinodes 
at  distances  Z/4,  3^/4,  etc.,  from  the  interface  ;  and  the  mag- 
netic antinodes  have  the  positions  of  the  electric  nodes  at  dis- 
tances o,  L/2,  L,  etc.,  from  the  interface.  Thus  the  magnetic 
intensity  is  a  maximum  or  minimum  where  the  electric  intensity 
is  zero,  and  the  electric  intensity  is  a  maximum  or  minimum 
where  the  magnetic  intensity  is  zero. 

(2)  Reflection  from  a  perfect  insulator  with  infinite  indue  tivity. 
In  this  case  the  magnetic  intensity  at  the  interface  must  be  zero, 
since  otherwise  the  induction  (B  =  fjiH)  in  the  dielectric  with  infi- 
nite inductivity,  and  the  "  magnetic  current,"  *  would  be  infinite 
except  at  two  instants  in  every  period.  Therefore  the  magnetic 
intensities  of  the  incident  and  reflected  wave  trains  must  be  equal 
and  opposite  at  the  interface.  Hence,  if  we  use  the  nomenclature 
of  (i),  we  have 

HI  =  Apjiin  -  [cos  (nt  —  pz)  —  cos  (nt  +  pz)~\ 

(  47  ) 
=  2Apj  fjLn  •  sin  27rt/  T-  sin  27rzj  L 


and  El=A  [cos  (nt  -  pz)  +  cos  (nt  + 

=  2  A  COS  27T//  T-  COS  ZTTZIL 

Thus  the  interference  of  the  two  trains  of  waves  produces  a 
system  of  standing  waves  in  which  the  magnetic  nodes  and  the 
electric  antinodes  are  located  at  the  interface  and  at  distances 
LJ2tLt  etc.,  therefrom,  while  the  magnetic  antinodes  and  electric 
nodes  are  located  at  distances  Z-/4,  3-£/4»  5^/4,  etc.,  from  the 
interface.  Thus  the  nodes  in  this  case  occupy  the  positions  of 
the  antinodes  in  (i),  and  the  antinodes  the  positions  of  the  nodes 
in  (I). 

*  That  is,  the  rate  of  change  of  magnetic  flux,  by  analogy  with  dielectric  current. 


462  ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

15.  The  Flux  of  Energy  in  a  System  of  Standing  Waves.  For 
the  electromagnetic  energy  flux  density  at  the  time  t  across  a 
plane  distant  z  from  the  XY  plane,  (45)  and  (46),  or  (47)  and 
(48),  §  14,  give 


R  =  V  E^H2  =  A^pj  fjin  •  sin  4?r//  T  •  sin 

(49) 
=  cvA2  sin    TrtT  •  sin    .TrzL 


since  n*jp2  =  v2  =  i  /  pc. 

Thus  R  is  permanently  zero  at  all  points  for  which  sin  ^.irzjL 
=  o,  that  is  at  all  electric  and  magnetic  nodes  (or  antinodes). 
At  any  point  between  an  electric  node  and  a  magnetic  node,  R 
goes  through  a  complete  cycle  of  positive  and  negative  values  in 
the  time  TJ2,  its  amplitude  being  greatest  at  points  for  which 
sin  ^.Trz/L  =  i,  that  is  points  half  way  between  electric  and  mag- 
netic nodes. 

At  any  instant  R  has  opposite  signs  on  opposite  sides  of  any 
node,  and  also  on  similar  sides  of  successive  nodes.  During  one 
quarter  of  a  period  the  energy,  wholly  electrostatic  at  the  start, 
streams  from  the  electric  antinodes  (magnetic  nodes)  toward  the 
electric  nodes  (magnetic  antinodes),  being  completely  trans- 
formed into  magnetic  energy  at  the  end  of  the  quarter  period. 
During  the  next  quarter  period  the  energy,  wholly  magnetic  at 
the  beginning,  streams  from  the  magnetic  antinodes  toward  the 
electric  antinodes,  being  completely  reconverted  into  electro- 
static energy  at  the  end  of  the  quarter  period.  The  energy 
density  has  now  everywhere  the  same  value  as  a  half  period 
earlier,  but  the  sign  of  the  electric  intensity  is  everywhere  oppo- 
site. During  the  next  half  period  the  same  energy  transfer  and 
transformations  occur,  and  at  its  close  the  electric  intensity  and 
the  energy  density  have  the  same  values  as  at  the  beginning  of 
the  period. 

16.  The  Propagation  of  a  Plane  Simple  Harmonic  Wave  Train  in 
a  Conducting  Medium  Containing  no  Other  Electromagnetic  Field 
than  That  of  the  Wave  Train.  In  this  case  we  have 


THE   FLUX   OF    ELECTROMAGNETIC    ENERGY.          463 

i=kE+  cdEjdt  =  curl  H  (50) 

(51) 


from  which  the  six  component  equations  can  be  easily  written 
down. 

From  these  equations,  by  a  process  exactly  analogous  to  that 
carried  out  in  §  1  1  ,  we  obtain  for  the  simple  case  in  which  E2  = 
£3  =  o,  and  E^  —  E  is  independent  of  x  and  y, 

k^dEJdt  +  pcd2^  I  dt2  =  d^EJdt?  (52) 

and  a  similar  equation  for  H2,  the  other  components  of  H  being 
zero. 

To  solve  (52)  for  the  simple  case  of  a  harmonic  wave  train  of 
given  period  7",  or  given  wave-length  L,  progressing  in  the  posi- 
tive direction  of  Z  with  the  velocity  v  (to  be  determined),  assume 

El  =  E=  Ae~mz  cos  2K/L-(vt  —  z)  (53) 

For  the  sake  of  brevity  put 

n  =  pv  =  2?r/  r=  2TTV  I  L  (54) 

then  (53)  becomes 

El  =  E=  Ae~ms  cos  (nt  —  pz)  (55) 

The  damping  factor  e~ma  is  inserted  on  account  of  the  dissipat- 
ing effect  of  resistance,  m  being  a  quantity  to  be  determined. 

Substituting  for  El  in  (52)  its  value  as  given  by  (55),  and 
equating  to  zero  separately  the  coefficients  of  sin  (nt  —  pz)  and 
cos  (nt  —  pz)  in  the  resulting  equation,  we  obtain  as  the  condi- 
tions that  (55)  may  be  a  solution  of  (52), 

—  n2fjic  —  m2  -f  pz  =  o 
and  —  fink  +  2mp  =  o 

Hence       m=n{—  pc/2  +  [(/«:/  2)2  +  (^  /  2nf]^  (56) 

and  p=n{ncl2+  [(^/  2)2  +  (^  /  2n)^}*  (57) 

both  m  and  p  being   positive  quantities,  since    the    waves  are 
damped  and  since  they  progress  in  the  positive  direction  of  z. 


464          ELEMENTS   OF    ELECTROMAGNETIC    THEORY. 
From  (54)  and  (57),  the  velocity  of  the  waves  is 

v=  «//=  I  /  {pc I  2  +  [(pc  I  2f  +  (pkJ2nf\^       (58) 


being  thus  a  function  of  n  as  well  as  of  ft,  c,  and  k. 
The  total  magnetic  intensity  is 


(59) 
cos  (rit  —pz)  +  m  sin  (»/  —       ~ 


If  we  put 

/  =  N  cos  0  and  m  =  N  sin  0 
we  have 

_V=(«2+/)i 
and 

0=  tan-1^// 

By  means  of  these  equations  (59)  may  be  written 

H2  =  A(m2  +  /)*  /  /**  '  cos  (»'  -p*-0)  (60) 


For  good  conductors,  such  as  liquid  electrolytes  and  metals, 
the  ratio  (pc  j  2)  j  (pk  j  2n)  =  «z/£  is  exceedingly  small  except  for 
enormous  values  of  n.  Thus,  even  for  so  great  a  value  of  n  as 
io6,  cnjk  for  common  aqueous  solutions  of  salts  and  acids  is  of 
the  order  io~3,  and  for  metallic  conductors  is  too  small  to  have 
been  detected  by  experiment.  For  good  conductors,  therefore, 
we  may  write  without  sensible  error,  except  for  enormous  values 

of  n, 

m  =/  =  n(pkJ2n)*  =  (/*£#/  2)* 
and 

6  =  tan-1  m\p  =  tan-1  I  =  Tr/4  (61) 

(55)  and  (59)  thus  become 

El  =  Ae-(^V**  cos  n\t-  (ii,k\2n}z\  (62) 


and 

ff  =  A(kl^e-^kn^z  cos  {n\t  -  (^/2n)h]  -  TT  /  '4}    (63) 


THE    FLUX   OF    ELECTROMAGNETIC    ENERGY.          465 

The  velocity  of  the  wave  train  in  the  conductor  is 

v  =  njp  =  (2n/pfy  (64) 


The  relations  between  E^  =  £,  H2  —  ff,  and  z  at  the  time 
/=  o,  are  similar  to  the  relations  between  q,  /,  and  /,  Fig.  1  19. 

At  a  distance  z  from  the  origin  the  amplitudes  of  the  electric 
and  magnetic  intensities  are  less  than  their  amplitudes  at  the 
origin  in  the  ratio  C****&*  to  i.  The  distance  i/m=  (2.  //*£#)*, 
in  which  the  amplitude  of  either  wave  is  reduced  to  I  fe  of  its 
value  at  the  origin,  is  called  the  relaxation  distance  for  the  given 
medium  and  the  given  value  of  n.  The  distance  in  which  the 
amplitude  of  either  intensity  falls  to  any  fraction  of  its  value  at  the 
origin  is,  like  the  relaxation  distance  I  /m,  inversely  proportional 
to  IJL,  k,  and  n.  Thus  if  JJL,  kt  or  n  is  very  great  the  intensity  of 
the  waves  falls  off  very  rapidly.  If  either  ky  JJL,  or  n  is  infinite, 
that  is  if  the  conductor  is  a  perfect  conductor,  its  inductivity  in- 
finite, or  the  frequency  of  the  waves  infinite,  all  ideal  cases,  the 
electromagnetic  disturbance  does  not  enter  the  conductor  at  all. 
Thus  a  perfect  conductor  or  a  medium  of  infinite  inductivity 
would  form  a  perfect  electric  and  magnetic  screen  in  either  a 
static  or  a  variable  electric  or  magnetic  field. 

For  copper,  when  n  =  2ir  x  100,  m  =  7r/2  approximately. 
In  this  case  the  amplitudes  of  the  intensities  are  reduced  to  the 
fractions  0.208,  0.043,  an^  less  than  1/500  the  origin  values  at 
the  distances  I,  2,  and  4  cms.,  respectively,  from  the  origin. 
When  n  —  2?r  x  1,000,000,  m  =  SOTT  approximately,  and  the 
amplitudes  are  reduced  to  less  than  1/6,000,000  the  origin 
values  in  going  a  distance  of  I  mm. 

In  the  case  of  iron,  if  //.=  1000,  m  =  20,  approximately,  when 
n  =  27r  x  100.  The  amplitude  of  either  intensity  falls  off  to 
about  thirteen  hundredths  and  one  twenty-thousandth  part  of 
the  value  at  the  origin  in  traversing  the  distances  I  mm.  and 
5  mm.,  respectively.  When  n  =  2ir  X  1,000,000,  m  —  2000  ap- 
proximately, and  the  amplitudes  fall  off  in  i/io  mm.  to  about 
one  five  hundred  millionth  part  of  their  values  at  the  origin. 


466          ELEMENTS   OF   ELECTROMAGNETIC   THEORY. 

These  examples  are  given  by  J.  J.  Thomson,  Elements  of  the 
Mathematical  Theory  of  Electricity  and  Magnetism,  p.  418. 

17.  The  Propagation  of  an  Electromagnetic  Field  into  a  Con- 
ducting Cylinder.  The  magnetic  field  of  a  long  circular  solenoid 
traversed  by  a  steady  current  is  described  in  §  20,  XII.  The 
electric  field,  if  the  resistance  of  the  solenoid  is  small,  is  weak, 
and  the  only  flux  of  energy  into  the  solenoid  is  the  flux  develop- 
ing the  Joulean  heat  in  the  wire.  In  what  follows  the  resistance 
of  the  solenoid  will  be  supposed  very  small  and  its  counter  e.m.f. 
negligible  in  comparison  with  the  e.m.f.  of  induction.  The  radius 
of  the  solenoid  will  be  denoted  by  a. 

If  the  magnetic  flux  through  the  coil  varies,  that  is  if  tubes  of 
magnetic  induction  move  outwards  or  inwards,  an  electric  field 
will  be  developed  within  and  without  the  solenoid.  The  lines  of 
electric  intensity  will  be  circles  centered  on  the  axis  in  planes 
perpendicular  thereto,  and  the  e.m.f.  around  any  circle  of  radius  r 
will  be  given,  in  magnitude  and  direction,  by 


where  dQ  jdt  is  the  rate  at  which  magnetic  flux  in  the  positive 
direction  crosses  the  circle  inwardly,  or  by 


where  u  is  the  velocity  of  the  tubes  of  magnetic  induction  at 
the  circle  of  radius  r.  The  electric  intensity  is  always  zero  on 
the  axis. 

The  energy  flux  density,  whose  direction  coincides  with  the  di- 
rection of  motion  of  the  electric  and  magnetic  tubes,  is  R  =  VEH, 
which  is  always  radial,  toward  or  from  the  axis. 

If  now  an  alternating  e.m.f.  acts  upon  the  coil,  the  electric  and 
magnetic  inductions  will  be  propagated  inwards  and  outwards 
alternately,  the  direction,  as  well  as  the  direction  of  motion,  of 
each  being  reversed  once  every  half  period. 

If  the  period  of  the  alternation  is  large,  so  that  the  distance 
traversed  by  the  tubes  of  induction  during  one  period  is  great  in 


THE   FLUX   OF   ELECTROMAGNETIC   ENERGY.         467 

comparison  with  the  radius  of  the  solenoid  and  the  radii  of  all 
circles  of  electric  intensity  considered  (all  supposed  small  in  com- 
parison with  the  length  of  the  solenoid),  the  magnetic  induction 
will  have  sensibly  the  same  value  throughout  the  solenoid  at  any 
instant.  The  electric  intensity  at  a  distance  r  from  the  axis  will 
be  approximately 


E=  - 
if  r  is  less  than  a  ;  and 


if  r  is  greater  than  a  ;  <£>  being  total  flux  in  the  positive  direction 
through  the  solenoid  at  the  time  t. 

If  the  period  of  the  alternation  is  small,  so  that  the  distance 
traversed  by  the  tubes  during  one  period  is  of  similar  magnitude 
to  that  of  the  radius  of  the  solenoid,  then  the  magnitude  and 
direction  of  both  intensities  will  vary  with  r. 

If  in  addition  the  core  of  the  solenoid  is  a  conductor,  or  if  it 
exhibits  hysteresis,  or  both,  as  when  made  of  iron,  then  the 
e'nergy  of  the  tubes  will  be  partly  dissipated  during  their  propa- 
gation in  the  core.  Hence,  since  the  direction  of  each  intensity 
is  periodically  reversed,  the  amplitude  of  the  magnetic  intensity 
as  well  as  that  of  the  electric  intensity,  will  steadily  diminish  as 
the  axis  is  approached. 

If  the  radius  of  the  core  is  great,  or  the  curvature  of  its  surface 
small,  the  law  of  the  diminution  of  the  intensities  with  the  dis- 
tance from  the  surface  is  approximately  the  same  as  that  deduced 
for  a  conductor  traversed  by  plane  waves,  §  16,  the  ratio  of  the 
amplitude  of  either  intensity  at  the  distance  z  from  the  surface  to 
its  surface  value  being  approximately  e-(^kni^z  j  i. 

Precisely  the  same  form  of  reasoning  applies  to  the  propaga- 
tion of  an  alternating  electromagnetic  field  into  the  cylindrical 
conductor  of  §  3. 

The  same  form  of  reasoning  also  applies  to  a  cylindrical  con- 
ductor (§  6)  to  whose  surface  the  electric  intensity  is  not  parallel, 


468    ELEMENTS  OF  ELECTROMAGNETIC  THEORY. 

since  the  parallel  component  only  is  concerned  in  the  propagation 
of  energy  into  the  conductor. 

Thus  a  rapidly  alternating  current  is  not  distributed  uniformly 
throughout  the  conductor,  but  is  more  or  less  concentrated  near 
its  surface.  This  increases  the  resistance  of  the  conductor,  and 
decreases  its  inductance,  the  former  being  greater  the  less  the 
area  of  the  section  across  which  the  current  flows,  and  the  latter 
being  less  the  thinner  and  farther  from  the  axis  the  walls  of  the 
tube  through  which  the  principal  part  of  the  current  now  flows 

[§2I,  (2),   XIII.]. 

In  the  case  of  a  very  thin  wire,  all  points  of  the  surface  are 
very  near  to  the  axis,  hence  both  the  above  effects  are  small. 

18.  The  Propagation  of  Waves  Along  Wires.  Since  the  elec- 
tromagnetic waves  discussed  in  §§11-15  are  propagated  un- 
changed at  right  angles  to  the  intensities,  it  is  clear  that  the 
results  there  obtained  hold  good  for  any  plane  plane  polarised 
wave,  whether  the  wave  front  is  infinite  or  not  and  whether  the 
direction  of  E  (as  well  as  that  of  //)  is  the  same  for  all  parts  of 
the  wave  front  or  not. 

Thus  they  apply  to  plane  waves  propagated  between  two 
parallel  perfectly  conducting  planes,  or  to  waves  propagated 
along  two  parallel  perfectly  conducting  cylinders  concentric  like 
those  of  §§  22,  XIII.,  and  I,  or  side  by  side  like  those  of  §  24, 
XIII.  That  the  electric  and  magnetic  intensities  of  these  systems 
are  perpendicular  at  any  point  in  the  case  of  electric  waves  as  in 
the  case  of  a  steady  current  is  apparent  from  previous  discus- 
sions without  reference  to  (32)  or  (33),  §  12. 

If  the  resistance  of  the  conductors  is  not  zero,  but  small,  the 
relations  deduced  in  the  articles  referred  to  above  will  apply 
approximately.  Thus  electric  waves  travel  along  wires  of  small 
resistance  surrounded  by  a  given  medium  with  approximately  the 
same  velocity  as  in  free  space  containing  the  same  medium. 

If  the  two  wires  are  joined  at  the  end  remote  from  the  oscil- 
lator by  a  large  plane  conductor  perpendicular  to  their  lengths, 


THE   FLUX   OF    ELECTROMAGNETIC    ENERGY. 


469 


or  even  by  bending  them  together,  a  system  of  standing  waves, 
resembling  that  of  §  14,  (i)  will  result,  with  an  approximate 
electric  node  and  magnetic  antinode  at  this  end. 

If,  on  the  other  hand,  the  wires  are  insulated  from  one  another 
at  this  end,  a  system  of  standing  waves,  resembling  that  of  §  14, 
(2)  will  result,  with  an  approximate  magnetic  node  and  electric 
antinode  at  this  end. 

19.  Mechanical  Analogue  of  an  Electromagnetic  Wave,  Waves 
in  Frictionless  Elastic  Media,  Consider  a  plane  transverse  wave 
traversing  an  infinite  elastic  medium  in  the  positive  direction  of  Z. 
Let  the  displacement  of  the  medium  take  place  parallel  to  the  Y 


Fig.  145. 

axis.  Then  at  the  time  /  while  the  disturbance  is  crossing  any 
plane  distant  s  from  the  XY  plane,  every  point  of  the  medium  in 
this  plane  will  be  shifted  in  the  same  direction  and  through  the 
same  distance,  yy  from  its  equilibrium  position. 

Fig.  145  shows  a  section  parallel  to  the  YZ  plane  of  a  portion 
of  the  medium  in  its  undisturbed  state  AB,  and  in  a  disturbed 
state  A'Bf  at  the  time  t  while  a  wave  is  passing.  Every  infini- 
tesimal parallelepiped  with  its  sides  parallel  to  the  coordinate 
planes  which  is  bounded  on  the  sides  parallel  to  XY  by  planes 
distant  z  and  z  -f  dz  from  the  XY  plane  is  shifted  and  sheared 


470          ELEMENTS   OF   ELECTROMAGNETIC    THEORY. 

precisely  like  the  parallelepiped  abed,  which  is  shifted  and 
sheared  into  the  parallelepiped  a'  b'  c'  d' . 

Let  the  shear  (equal  to  the  angle  between  ab  and  a'b')  at  the 
plane  z  be  denoted  by  D.  Then 

D  =  -  dyjdz  (65) 

the  negative  sign  being  chosen  since  D,  which  is  a  vector  per- 
pendicular to  ab  and  a'b1 ,  is  positive,  as  in  the  case  represented 
in  the  figure,  when  a  right-handed  screw  rotating  from  ab  to  a'b' 
would  move  in  the  positive  direction  of  X  (up  from  the  plane  of 
the  paper  in  the  figure). 

Let  the  area  of  each  face  of  the  parallelepiped  parallel  to  the 
XY  plane  be  denoted  by  dS\  and  let  the  modulus  of  rigidity,  or 
shear  modulus,  of  the  medium  be  denoted  by  n  =  I  jct  c  being 
the  shear  permittivity.  Then,  if  E  denotes  the  shearing  stress 
in  the  plane  z, 

E=nD=  —  ndyjdz  =  —  I  jc  •  dyjdz  (66) 

The  shearing  force  upon  the  face  a' c'  is  therefore 

EdS=nDdS 
and  that  upon  the  face  b' d'  is 

-(£+  dEjdz  dz)dS  =  -n(D  +  dDjdz  dz)dS 

the  force  being  positive  when  directed  in  the  positive  direction  of 
the  Faxis.  The  total  force  upon  the  parallelepiped  a'bf c' d'  is 

therefore  JKIJ   j  vc 

—  a h  I  dz  azas 

Let  the  density  of  the  medium  be  denoted  by  p.  Then,  since 
the  volume  of  the  parallelepiped  is  dz  dS,  its  mass  is 

ndzdS 

Let  the  velocity  of  the  parallelepiped  be  denoted  by  H  (posi- 
tive when  in  the  positive  direction  of  F).  Then 

ff=  dy\dt  (67) 


THE    FLUX   OF    ELECTROMAGNETIC    ENERGY.          471 

Hence,  by  the  second  law  of  motion, 

iriz  dS  dHjdt  =  -  dEjdz  dz  dS 
whence 

fidHjdt  =  -  dEjdz  (68) 

By  differentiating  (66)  with  respect  to  t  we  obtain 

cdEjdt  =  dD\dt  =  -  d\dt*  (dy\dz) 

=  -  dldz  -  (dyldt)  =  -  dHjdz       ^ 

By  differentiating  (69)  with  respect  to  t  and  (68)  with  respect 
to  z  and  combining  the  resulting  equations  we  obtain 

c^Efdt2  =  d2Ejdz2  (69) 

or  cpd2Djdt*  =  dzDjdt*  (70) 

By  differentiating  (69)  with  respect  to  z  and  (68)  with  respect 
to  t  and  combining  the  resulting  equations  we  obtain 

cpd^Hldt*  =  d^Hldz*  (71) 

or  cpd2Bjdt2  =  d2Bjdzz  (72) 


Equations  (69^(72)  are  identical  with  equations  (i7)-(2o), 
and  show  that  all  the  results  of  §§11-15  apply  also  to  the 
dynamical  waves  here  considered,  the  shear  permittivity  and 
density  of  an  elastic  medium  being  substituted  for  the  electric 
permittivity  and  magnetic  inductivity  of  a  dielectric,  shearing 
stress  and  shear  for  electric  intensity  and  displacement,  and  lin- 
ear velocity  and  momentum  per  unit  volume  for  magnetic  inten- 
sity and  induction. 

By  introducing  internal  friction,  the  analogy  may  be  readily 
extended  to  the  damped  electromagnetic  waves  of  §  16. 

20.  The  Stresses  in  an  Electromagnetic  Wave.  Electromag- 
netic Radiation  Pressure.  At  every  point  in  an  electromagnetic 
wave  there  is  a  pressure  normal  to  the  plane  containing  the  elec- 
tric and  magnetic  intensities,  that  is  normal  to  the  wave  front, 


472          ELEMENTS    OF    ELECTROMAGNETIC    THEORY. 

equal  to  the  sum  of  the  electric  and  magnetic  pressures,  or  the 
sum  of  the  electric  and  magnetic  energy  densities,  at  the  point, 
by  §§40-41,  I.,  and  §  18,  XI.  Thus,  if/  denotes  this  pressure, 

/=^£2+i^  (73) 

In  a  single  wave,  or  train  of  waves,  ^cE2  =  \^H2  at  any  point, 
and,  as  seen  in  all  the  electromagnetic  waves  considered  above, 
the  electric  and  magnetic  intensities  at  any  point  are  perpendicu- 
lar to  one  another.  Hence  the  electric  tension  in  the  direction  of 
the  electric  intensity  is  just  neutralised  by  the  magnetic  pressure 
perpendicular  to  the  magnetic  intensity,  and  the  magnetic  tension 
in  the  direction  of  the  magnetic  intensity  is  just  neutralised  by  the 
electric  pressure  perpendicular  to  the  electric  intensity.  Thus 
the  pressure  p  normal  to  the  plane  of  the  intensities  is  the  total 
(dynamical)  stress  in  the  wave. 

In  wave  systems  in  which  at  any  point  %cE2  is  not  equal  to 
J/i//2,  as  the  systems  of  §  14,  there  is  in  addition  to  the  normal 
pressure  /,  a  tension  \cE2  —  \pH2,  or  a  pressure  \pPP  —  %c£2, 
parallel  to  the  electric  intensity,  and  a  tension  \pH2  —  \cE2,  or 
a  pressure  %cE2  —  ^^H2,  parallel  to  the  magnetic  intensity. 

If  electromagnetic  waves  in  a  given  dielectric  (i)  are  incident 
normally  upon  the  interface  separating  this  dielectric  from  another 
medium  (2),  at  the  surface  of  which,  or  within  which,  the  inten- 
sities, and  therefore  the  pressures,  are  reduced  to  zero  (by  total 
reflection  from  the  interface,  partial  reflection  and  partial  absorp- 
tion, or  total  absorption),  the  interface  will,  in  accordance  with 
what  precedes,  experience  a  force  directed  toward  medium  (2) 
and  equal  to  p  x  the  area  of  the  interface  exposed  to  the  waves. 
If  the  waves  are  partially  transmitted  through  medium  (2),  emerg- 
ing at  a  second  interface,  the  total  pressure  upon  medium  (2)  in 
the  direction  of  the  propagation  of  the  incident  waves  is  equal  to 
the  difference  between  the  values  of  /  at  the  two  interfaces. 

If  medium  (2)  is  a  perfect  conductor,  the  waves  are  totally  re- 
flected, the  electric  intensity  at  the  interface  is  zero,  and  the  mag- 
netic intensity  at  the  interface  is  twice  the  magnetic  intensity  of 


THE    FLUX    OF    ELECTROMAGNETIC    ENERGY.          473 

the  incident  wave,  that  is,  2H  cos  nt,  if  H  cos  nt  denotes  the 
magnetic  intensity  at  the  interface  of  the  incident  wave.  Thus 
the  radiation  pressure  upon  the  interface  at  the  time  /  is 

p  =  |/i(2H  cos  nt)2  =  2/xH2  cos2  nt  (74) 

and  the  mean  value  of  the  pressure  during  a  complete  period  is 
Q))  =  2/u.H2  x  mean  value  of  cos2  nt 

==  2/iH2  x  mean  value  of  (  J  -f  J  cos  2nt)  =  /-iH2 

If  medium  (2)  is  a  non-conductor  with  infinite  inductivity,  the 
waves  are  totally  reflected,  the  magnetic  intensity  at  the  interface 
is  zero  permanently,  and  the  electric  intensity  there  is  twice  the 
electric  intensity  of  the  incident  wave,  that  is  2E  cos  nt,  if  E  cos  nt 
denotes  the  electric  intensity  of  the  incident  wave  at  the  inter- 
face. Thus  the  pressure  upon  the  interface  at  the  time  /  is 

p'  =  \c(2&  cos  nt)2  =  2rE2  cos2  nt  =  p  (76) 

and  the  mean  value  of/'  during  a  complete  period  is 

j>')  =  '*'-(j»)  (77) 

If  the  energy  of  the  incident  wave  is  totally  absorbed  by  me- 
dium (2),  there  is  no  reflected  wave,  and  the  pressure  upon  me- 
dium (2)  is 

p"  =  1<E  cos  nff  +  J/*(H  cos  nff 

(78) 
=  J(VE2+  /*H2)  cos2  nt  =  \p'  =  \p 

The  mean  value  of  the  pressure  during  a  complete  period  is 

(79) 


For  experimental  investigations  confirming  in  a  striking  man- 
ner the  theory  of  radiation  pressure,  developed  independently 
and  in  different  ways  by  Maxwell  and  Bartoli,  see  P.  Lebedew, 
Ann.  der  Physik,  Vol.  6,  p.  433,  1901  ;  and  especially  Nichols 
and  Hull,  Astrophysical  Journal,  Vol.  17,  p.  315,  1903. 

For  the  theory  of  vibration  pressure  in  general,  see  Lord 
Rayleigh,  Philosophical  Magazine,  Vol.  3,  p.  338,  1902. 


INDEX. 


Absorption,  electric  176 

Alternating  currents,   etc.  382,  390,  396, 

399 

resistance  of  conductor  for,  466 
Ampere's  law  288,  298 
Analogues,  dynamical  (see  dynamical} 
Angle 

of  lag  38 1,  383 

of  lead  38 1,  383 
Anion  228,  232 
Anode  203 
Apparent  charge  142 
Axis 

of  a  doublet  102 

of  a  magnet  266 

Balance,  electrodynamic  363 
Ballistic 

galvanometer  322 

method  of  measuring  induction,  etc. 

369,  402 
Bridge 

Kelvin's  double  227 

Wheatstone's  223 

Capacity 

of  an  electrical  system  28 
of  a  conductor  56 

comparison  of,  with  :  another  capa- 
city   135,    136,    330,    331,    413; 
inductance    404  ;    mutual   induct- 
ance 409  ;  resistance  329,  411 
Circuitation 

first  law  of  303,  358,  359,  430 
second  law  of  340,  358 
Circular  conductors,  magnetic  field  sur- 
rounding 300,  301 
Charge,  electric 

apparent  or  fictitious  142 
residual  176 


Charge,  electric 
true  I,  6,  7 
measurement  of  324 
Coefficient 

of  capacity  47 

of  induction  (electrostatic)  47 

of  induction    (electromagnetic)  333 

of  mutual  induction  333 

of  potential  or  voltage  46 

of  self  induction  332 
Coercive  force  or  intensity  371 
Coherer  452 
Coil 

flux  through  a  332 

revolving  389 

various  forms  of  ( see/fe/<r/r,  magnetic} 
Concentration  234 
Condensance  383 
Condenser  29 

cylindrical  66,  126 

discharge  of  54,  375,  379 

parallel  plate  70,  123,  124,  148 

spherical  58,  124,  150 

standard  124-126 

systems  133,  134 

various  forms  of  (seeyfofafr,  electric'] 
Conductance  203 
Conduction 

electric  2,  37 

electrolytic  228 

electron  theory  of  244 

metallic  228,  244 

current  199 
Conductivity 

electric  205 

electrolytic  238 

molecular  238 
Conductors 

electric  2 
475 


4/6 


INDEX. 


Conductors,  magnetic  274 
Convection  current  199 

experiments  on  472 

fictitious  429 

magnetic  fields  of  424-427 
Convergence  of  a  vector  23 

Cartesian  expression  for  25 
Coulomb's  law 

in  electrostatics  8 

in  magnetostatics  267 
Curl  of  a  vector  340 

Cartesian  expression  for  439 
Current,  electric  conduction  200 

convection  199,  201,  427 

dielectric  or  displacement  200,  424 

measurement  of  229,  319,  327,  329, 

363,  364 

total  electric  431 

magnetic  461 
Curvature,  relation  of  to  electric  surface 

density  35 
Cylinder 

electric  field  of  isolated  65 

electromagnetic  field  of  437,  443 

in  uniform  field   163  (electric),  291 
(magnetic) 

magnetic  field  of  isolated  303 
Cylinders,  coaxial 

electric  field  of  65 

electromagnetic  field  of  426,  429,  433 

magnetic  field  of  305 
Cylindrical 

condenser  66,  126 

shell  in  uniform  field  166 

Decrement,  logarithmic  381 
Density 

electric  surface  and  volume  23 

energy  (see  energy} 

fictitious  142 

magnetic  surface  and  volume  276 
Deprez-D' Arsonval    galvanometer    316, 

324 
Diagrams 

Maxwell's,  of  electric  and  magnetic 

fields  63,  68,  72  (see  *\sa  fields} 
thermoelectric  256,  260 


Diamagnetic  substances  366 
Dielectric  I 

constant  8 

current  200,  424 
Dimensions  422 
Discharge 

by  successive  contacts  54 

induction,  through  a  circuit  342 

oscillatory  379 

unidirectional  374,  375 
Displacement 

electric  II 

current  200,  424 
Dissociation  232 

ratio  238 
Divergence  of  a  vector  23 

Cartesian  expression  for  25 
Doublet,  electric 

line  87 

point  76,  102,  103 

Dynamical  and  electrical  analogues  II, 
12,  30,  177,  202,  213,  220,  343,  344, 
385,  469,  473 

Earth,  electric  field  surrounding  62 

Electrets  181-191 

Electrisation 

intensity  of  145 

intrinsic  181 
Electrodes  202 
Electrodynamic  balance  363 
Electrodynamometer  327,  363,  364 
Electrolysis  228 
Electromagnetic  induction  333 
Electrometers 

absolute  126,  128 

quadrant  130 
Electromotive  force  14 

impressed  and  intrinsic  213,  217 

induced  336,  339 

motional  334 

thermal  246 

comparison  of,  with  :  another  e.m.f. 
126-130,  328;  current  X  resist- 
ance 328 

Electromotive  forces,  superposition  of  27, 
217 


INDEX. 


477 


Electrons  244 
Electrostriction  170 
Energy 

electric  30,  44 

electric,  mechanical,  and  change  of 
configuration  51 

electrokinetic  or  magnetic  275,  342, 

347 
electrokinetic,       mechanical,       and 

change  of  configuration  360 
density 

electric  31 

electrokinetic  or  magnetic  274, 

345 

flux  of  electromagnetic  440 
mutual  electrokinetic  348 
Equipotential  surface 
electric  1 6 
magnetic  273 

Ferromagnetic  substances  366 
Fictitious 

charges  142 

convection  current  429 
Field 

electric  10 

magnetic  269 
Fields 

electrostatic,    various    57-167,    182- 
191 

magnetostatic,  various  280-281 

magnetic  (of  currents),  various 288- 

3l6>  35T-357 
electromagnetic,     various    424-42?^ 

429,  433,  437,  443-469 
equilibrium  of  22,  26,  32-44,  274 
superposition  of  10,  26,  72,  271,  274 

Flux 

electric  17 

of  electromagnetic  energy  440 
magnetic  271,  365 
magnetic,  through  a  coil  332 

Force 

coercive  371 

electric  and  magnetic  (see  intensity} 
on  various  conductors  and  insulators 
(SKS  fields} 


Galvanometers  316-327 
Gauss's  theorem 

in  electrostatics  17,  22,  139  (gener- 
alised) 

in  magnetostatics  27 
Gaussage  273 
Gram  atom  233 

ion  233 

molecule  233 

Hollow  conductors,  experiments  with  4 
Hysteresis 

dielectric  179 

magnetic  370 

Images 

electric  43 

various  (see fields,  electric} 
electric,  by  inversion  114 
geometrical  97 
Impedance  383 
Impressed 

electric  intensity  181,  218 
electromotive  force  213 
Inductance  332,  349 

and  linear  dimensions  349 
and  number  of  turns,  348 
comparison  of,  with  :  another  induct- 
ance  408  ;  capacity  404  ;  mutual 
inductance  410 ;  resistance  413 
Inductances 

in  series  350 
standard  350 
Induction 

coefficient  of  (see  coefficient} 
electric  n 

electrification  by  3,  37 
electromagnetic  333 
magnetic  270,  365 
remanent  371 
residual  371 
measurement  of  magnetic 

by  the  ballistic  method  369,  402 
by  the   magnetometric   method 

367 

Inductive  circuits  375 
Inductivity,  magnetic  268,  365 
Insulators  2 


INDEX. 


Intensity 

coercive  371 
electric  10 

impressed  218 

induced  334,  336 

intrinsic  218 

motional  333 
magnetic  269 

due  to  currents  297 

induced  424 

intrinsic  428 

motional  428 

measurement  of  282,  402 

comparison  of,  with  another  mag- 
netic intensity  285 
Intrinsic 

displacement,  etc.  181 
electric  intensity  216 
electromotive  force  213 
magnetic  intensity  279,  428 
magnetisation  279,  371 
Inverse  squares,  law  of 
in  electrostatics  8,  6 1 
in  magnetostatics  267 
Inversion 

electrical  113 
geometrical  no 
thermoelectric  252 
Ions  228,  232 

velocity  of  234 

Joule's  law  21 1 

Kathode  203 

Kation  228,  232 

Kirchhofi's  laws  202,  222,  431 

Lag,  angle  of  381,  383 
Laplace's  equation  25,  60,  67,  71 
Lead,  angle  of  381,  383 
Lenz's  law  338 
Leyden  29 
Line 

of  current  or  flow  202 

of  intensity,  induction,  etc.,  14,  271 

thermoelectric  253 

Lorenz's  method  of  determining  a  resis- 
tance 341 


Magnet  265 

permanent  279 

torque  on  282 
Magnetic  substances  366 
Magnetisation 

curves  365 

intensity  of  276 

measurement  of  367,  369 

intrinsic  371 

work  done  in  359 
Magnetism 

quantity  of  268,  278 
j  .         remanent  371 

residual  371 
Magnetometer  283 
Magnetometric  method  366 
Magnetomotive  force  273 

induced  424 

Mance's  method  of  measuring  the  resis- 
tance of  a  generator  226 
Moment 

of  a  doublet  76,  88 

of  a  magnet  282,  285 
Motional 

electric  intensity  and  e.m.f.  333 

magnetic  intensity  and   m.m.f.  428 
Mutual  inductance  333,  349 

and  linear  dimensions  of  circuits  349 

and  number  of  turns  348 

comparison  of,  with  :  another  mutual 
inductance  401,  403  ;  a  capacity 
409  ;  an  inductance  410  ;  a  resis- 
tance 341,  400 

Neutral  temperature  252 
Non-inductive  circuit  374 

Ohm's  law 

for   a   homogeneous    conductor  203, 

206,  238 

for  a  steady  current  219 
for  variable  currents  374-379,  383, 

394-396 

general  forms  of  217,  218 
Oscillatory  discharge  379 

Parallel  cylinders 

electric  field  of  83 


INDEX. 


479 


Parallel  cylinders 

magnetic  field  of  306 
Paramagnetic  substances  366 
Peltier  effect  246 
Permeability  365 
Permeance  275 
Permittance  28 
Permittivity  8 

Permittivities,  comparison  of  192 
Piezoelectric  crystals  1 88 
Phase  difference  381,  383 
Plane 

infinite  charged  69 

in  presence  of  a  concentrated  charge 

116 
Planes 

intersecting  at  various  angles  107 

magnetic  field  between  parallel  312 
Poisson,  equation  of  25 
Poles 

electric  144 

magnetic  266,  281  (resultant) 
Pole  strength 

electric  144 

magnetic  268,  278 
Potential 

electric  16,  61,  147 

magnetic  273,  278 
Potential  difference 

electric  14 

magnetic  273 

single  263 
Poynting's  theorem  433-448,   458,  459, 

462,  466 
Pressure 

electric  32,  35 

magnetic  274 

electromagnetic  (radiation)  471 
Pyroelectric  crystals  187 

Radiation,  electric  448 
Radiator,  electric  449 
Ratio 

dissociation  238 

Hittorf's  235 

of  ES  and  EM  units  420 

of  transformation  398,  399 


Reactance,  383 
Refraction 

of  lines  of  electric  displacement  140 

of  lines  of  magnetic  induction  276 

of  stream-lines  208 
Reluctance  275 
Remanent  magnetism  371 
Residual 

charge  176 

magnetism  371 
Resistance  203,  213 

absolute  determination  of  213,  341, 
401  (see  also  comparisons  ff . ) 

to  alternating  currents  466 

comparison  of,  with  :  another  resis- 
tance 226,  227,  402,  404  ;  a  ca- 
pacity 329,  411;  an  inductance 
413  ;  a  mutual  inductance  400 

of  conductors  in  multiple  203 

of  conductors  in  series  204 

of  various  conductors  209 

specific  205 

temperature  coefficient  242 
Resistivity  205 
Resonance,  384 
Resonator  45 1 
Retentiveness  371 

Screens,  electric  4,  164,  166 

Seebeck  effect  246 

Self  induction  (see  inductance} 

Single  potential  difference  263 

Solenoid 

circular  316 
infinite  309 

inductance  and  energy  of  field  of  35 1 
Solenoidal  electrisation  146 
Specific 

heat  of  electricity  248 
inductive  capacity  192 
resistance  205 
Sphere 

conducting 

isolated  59 

in     presence     of    concentrated 

charge  96,  98,  116 
in  uniform  field  100 


INDEX. 


Sphere 

insulating,  in  uniform  field  158 
Spheres 

in  contact  118 

intersecting  at  right  angles  117 
Spherical 

condenser  (see  condenser] 

electret  (see  electrets} 

shell  (dielectric)  in  uniform  field  164 
Spheroid,  isolated  90 
Stream-lines  and  stream-tubes  202 
Strength  of  a  tube  of  induction 

electric  19 

magnetic  272 
Susceptibility 

electric  145 

magnetic  277 
Systems,  electrical 

with  inductance   and    capacity  373, 
391 

with  mutual  inductance  392,  396 

of  insulated  conductors  44-55 

Tension 

electric  32-35 

magnetic  274 
Thermal  effect,  reversible,  in  dielectrics 

168 

Thermal  e.m.f.s  246 
Thermocouple  246 
Thermoelectric 

diagram  2.0,  260 

line  253 


Thermoelectric 

power  250 
Thermoelement  246 
Thomson  effect  247 
Time  constant  375,  376 
Toroid,  field  and  inductance  of  312,  354 
Transformer  392 
Tubes  of  intensity,  etc. 

electric  14 

magnetic  271 


electromagnetic  systems  290,  416 
electrostatic  systems  290,  415 
practical  421 
ratio  of  ES  and  EM  420 
table  of  419 

Vector  product  287 

Vibrator  449 

Voltage  14 

Voltages,  superposition  of  27,  217 

Voltaic  cell,  reversible 
flux  of  energy  in  444 
von  Helmholz's  theory  of  261 

Waves,  electric  448 

along  wires  468 

in  non-conducting  dielectric  453,  458 

in  conducting  media  462 

reflection  of  460 

standing  460—462 

stresses  in  471 
Wheatstone's  bridge  223 


ELEMENTS  OF  PHYSICS 

FOR    USE    IN    HIGH    SCHOOLS 

BY 

HENRY   CREW,  PhJX 

Professor  of  Physics  in  Northwestern  University 

i2ino.     Cloth,    xiv  -f  347  pp.     Price,  $1.10. 

The  treatment  differs  from  other  elementary  books  on  the  same  subject 
in  that  it  is  more  consecutive.  The  aim  has  been  to  build  upon  the  average 
experience  of  a  student,  and  to  unify  the  discussions  of  Mechanics,  Sound, 
Heat,  Light,  and  Electricity  in  such  a  way  that  even  the  beginner  does  not 
feel,  in  passing  from  one  to  the  other,  that  he  is  undertaking  a  totally  new 
study.  By  this  plan  it  is  hoped  that  the  high-school  student  will  obtain  the 
soundest  and  most  economical  training,  whether  for  the  sake  of  liberal  cul- 
ture or  for  later  use  in  college  work,  engineering,  or  medicine.  The  treat- 
ment is  at  every  point  experimental  and  quantitative. 

TABLE    OF  CONTENTS 

INTRODUCTORY 

Chapter  I.  —  Motion.  Chapter  II.  — Simple  Harmonic  Motion.  Chap- 
ter III.  — General  Properties  of  Matter.  Chapter  IV.  — Special  Properties 
of  Matter.  Chapter  V. — Waves.  Chapter  VI.  —  Sound.  Chapter  VII. — 
Heat.  Chapter  VIII.  —  Magnetism.  Chapter  IX. — Electrostatics.  Chap- 
ter X. —  Electric  Currents.  Chapter  XI. —  Light.  Appendix  to  Chapter  IV. 


COMMENT 

' '  It  seems  to  me  that  heretofore  new  text-books  on  elementary  physics 
and  new  editions  of  old  ones  (with  some  few  exceptions),  have  been  new 
merely  in  that  they  appeared  in  new  covers  and  had  been  filled  out  a  little 
by  the  incorporation  of  a  few  new  and  remarkable  discoveries.  Professor 
Crew  has  written  a  new  book  from  beginning  to  end,  and  I  doubt  if  his 
method  of  treating  the  subject  could  be  improved  upon." 

— PROFESSOR  R.  W.  WOOD,  University  of  Wisconsin. 


THE    MACMILLAN    COMPANY 

66   FIFTH   AVENUE,    NEW  YORK 


<S  OUTLINES  OF  PHYSICS 

AN    ELEMENTARY    TEXT-BOOK 

BY 
EDWARD    L.   NICHOLS 

Professor  of  Physics  in  Cornell  University 

i2mo.    Cloth,    xi  +  452  pp.    Price,  $1.40 
Questions  to  same,  price  10  cents 

In  this  volume  the  author  has  outlined  a  short  course  in  physics  which 
should  be  a  fair  equivalent  for  the  year  of  advanced  mathematics  now 
required  for  entrance  to  many  colleges.  The  subject  is  divided  into  five 
parts  as  follows  : 

Part      I. — Mechanics. 

Part    II.  — Heat. 

Part  III.  — Electricity  and  Magnetism. 

Part  IV.  —  Sound. 

Part    V.— Light. 

Appendices. 

A  combined  class-book  and  laboratory  manual  which  is  logical  in  arrange- 
ment and  clear  in  its  statement  of  principles  and  descriptions  of  experiments. 


COMMENTS 

"Nichols's  'Outlines  of  Physics'  is  the  first  satisfactory  elementary 
physics  I  have  ever  seen,  after  searching  seven  years  for  one.  We  shall 
use  it  next  year. ' ' 

—  PROFESSOR  JAMES  BYRNIE  SHAW,  Illinois  College,  Jacksonville,  III. 

"  I  note  extreme  clearness  and  simplicity  of  explanation  in  the  text ;  all 
useless  details  are  omitted  and  the  author  aims  at  his  point  at  once,  so  that 
one  cannot  help  reading  ideas  instead  of  words.  Another  plan,  which  seems 
to  me  to  be  an  excellent  one,  is  the  placing  of  the  descriptive  text  before 
the  experiment  to  be  performed,  so  that  the  experiments  serve  to  verify  the 
author's  statements.  .  .  .  Good  judgment  is  shown  in  selecting  simple 
apparatus  for  performing  the  experiments.  As  an  all-around  up-to-date 
book  it  is  the  best  I  have  ever  seen." 

—  R.  WESLEY  BURNHAM,  High  School,  Gloucester,  Mass. 


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ELEMENTARY  LESSONS  IN  ELECTRICITY 
AND  MAGNETISM 

Efy  Professor  SILVANUS    THOMPSON 

f  First  Edition,  1881  ;  reprinted  1882  (2),  1883,  1884,  1885,  1886,  1887,  1889,  1890  (2),  1891   (2),  1892 
(3),  1894.     Second  Edition,  January,  1895 ;  reprinted  November,   1895,1897,1899.] 

New  Edition  Revised  Throughoiit  with  Additions. 
8vo.    Cloth,    xv  +  634  pp.    Price,  $1.40 

"From  beginning  to  end  the  subjects  are  judiciously  chosen,  admirably  dealt  with, 
and  logically  arranged,  forming  as  a  whole  what  is  unquestionably  the  standard  ele- 
mentary text-book  of  the  day.  We  do  not  say  it  is  the  best ;  we  go  further,  and  say 
it  is  the  only  book  we  can  honestly  recommend  to  the  junior  student." 

NATURE — "  Whoso  seeks  a  class-book  on  electricity  and  magnetism,  containing 
an  elementary  exposition  of  recent  work,  will  find  their  want  supplied  by  Professor 
Thompson's  lessons." 

A  PARTIAL   LIST   OF  ADOPTIONS 


University  of  California. 

Washington,  D.  C. 

Athens,  Ga. 

University  of  Illinois. 

Rose  Polytechnic  Institute,  Terre  Haute, 

Ind. 

University  of  Indiana. 
Purdue  University. 
Iowa  City,  la. 

University  of  Kansas,  Lawrence. 
Baldwin,  Kas. 

Center  College,  Danville,  Ky. 
Lexington,  Ky. 
Baltimore,  Md. 

Harvard  College,  Cambridge,  Mass. 
University  of  Michigan. 
Rolla,  Mo. 

Stevens  School,  Hoboken,  N.  J. 
Y.  M.  C.  A.,  Brooklyn,  N.  Y. 
Manual  Training  High  School,  Brooklyn. 
Boys'  High  School,  Brooklyn. 
Pratt  Institute,  Brooklyn. 


Commercial  School,  Buffalo,  N.  Y. 
Board  of  Education,  N.  Y.  City. 
Horace  Mann  School,  N.  Y.  City. 
Y.  M.  C.  A.,  New  York  City. 
Rochester,  N.  Y. 
Utica,  N.  Y. 

Clarkson  Memorial  School,  Potsdam,  N.Y. 
Rensselaer    Polytechnic    Institute,     Troy, 

N.  Y. 

Trinity  College,  Durham,  N.  C. 
Raleigh,  N.  C. 

Ohio  Wesleyan  University,    Delaware,  O. 
Pennsylvania   Military  Academy,  Chester, 

Pa. 

Temple  College,  Philadelphia. 
Erie,  Pa. 
Pittsburgh,  Pa. 
Clemson  College,  S.  C. 
Clarksville,  Tenn. 
University  of  West  Virginia. 
Y.  M.  C.  A.,  Richmond,  Va. 


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A  LABORATORY  MANUAL  OF  PHYSICS 
AND  APPLIED  ELECTRICITY 

ARRANGED   AND   EDITED    BY 

EDWARD    L.  NICHOLS,  B.S.,  Ph.D. 

Professor  of  Physics  in  Cornell  University 
IN  TWO  VOLUMES 

VoL  L  JUNIOR  COURSE  IN  GENERAL  PHYSICS 

BY 

ERNEST    MERRITT  AND   FREDERICK   J.    ROGERS 
Cloth.    $3.00 

Vol.  IL  SENIOR  COURSES  AND  OUTLINE  OF 
ADVANCED  WORK 

BY 

GEORGE   S.   MOLER,  FREDERICK    BEDELL,  HOMER  J.  HOTCHKISS, 

CHARLES   P.   MATTHEWS,    AND   THE   EDITOR 

Cloth.    Pp.444.    $3.25 

"  The  work  as  a  whole  cannot  be  too  highly  commended.  Its  brief  outlines  of  the 
various  experiments  are  very  satisfactory  ;  its  descriptions  of  apparatus  are  excellent ; 
its  numerous  suggestions  are  calculated  to  develop  the  thinking  and  reasoning  powers 
of  the  student.  The  diagrams  are  carefully  prepared,  and  its  frequent  citations  of 
original  sources  of  information  are  of  the  greatest  value." — Street  Railway  Journal. 

"The  work  is  clearly  and  concisely  written,  the  fact  that  it  is  edited  by  Professor 
Nichols  being  a  sufficient  guarantee  of  merit." — Electrical  Engineering. 


i,  THE  ELEMENTS  OF  PHYSICS 

By  EDWARD   L.  NICHOLS,   B.S.,  Ph.D. 

Professor  of  Physics  in  Cornell  University 
AND 

WILLIAM    S.  FRANKLIN,  M.S. 

Professor  of  Physics  and  Electrical  Engineering  at  the  Lehigh  University 
Complete  in  Three  Volumes.  (     Vol.     I.     Mechanics  and  Heat. 


Vol.  II.,  $1.90  net.  -j  II.     Electricity  and  Magnetism. 

Vols.  I.  and  III.,  each  $1.50  net.     (  III.     Sound  and  Light. 

The  ELEMENTS  OF.  PHYSICS  is  a  book  which  has  been  written  for  use  in  such  institu- 
tions as  give  their  undergraduates  a  reasonably  good  mathematical  training.  It  is 
intended  for  teachers  who  desire  to  treat  their  subject  as  an  exact  science,  and  who 
are  prepared  to  supplement  the  brief  subject-matter  of  the  text  by  demonstration, 
illustration,  and  discussion  drawn  from  the  fund  of  their  own  knowledge. 


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THE  ELEMENTS  OF  ALTERNATING 

YTDIST    CURRENTS  A   a^A- 

BY 

W.  S.  FRANKLIN  and  R.  B.  WILLIAMSON 

Second  edition,  revised.    8vo.    Cloth.    Price,  $2.50,  net. 

This  book  represents  the  experience  o  seven  years'  teaching  of  alternating  cur- 
rents, and  almost  every  chapter  has  been  subjected  repeatedly  to  the  test  of  class-room 
use.  The  authors  have  endeavored  to  include  in  the  text  only  those  things  which 
contribute  to  the  fundamental  understanding  of  the  subject  and  those  things  which 
are  of  importance  in  the  engineering  practice  of  to-day. 


CONTENTS 
CHAPTER  I. — Magnetic  flux.     Induced  electromotive  forcer.     Inductance.     Capacity. 

CHAPTER  IL — The  simple  alternator.     Alternating  e.m.f.  and  current.     The  contact 
maker. 

CHAPTER  III. — Measurements  in   alternating  currents.      Ammeters.       Voltmeters. 
Wattmeters. 

CHAPTER'  IV. — Harmonic  electromotive  force  and  current. 

CHAPTER  V. — Problem  of  the  inductive  circuit.      Problem  of  the  inductive  circuit 
containing  a  condenser.     Electrical  resonance. 

CHAPTER  VI. — The  use  of  complex  quantity. 

CHAPTER  VII. — The  problem  of  coils  in  series.     The  problem  of  coils  in  parallel. 

The  problem  of  the  transformer  without  iron. 
CHAPTER  VIII. — Polyphase  alternators.     Polyphase  systems. 
CHAPTER  IX. — The  theory  of  the  alternator.     Alternator  designing. 
CHAPTER  X. — The  theory  of  the  transformer. 

CHAPTER  XI.— Transformer  losses  and  efficiency.     Transformer  connections.     Trans- 
former designing. 

CHAPTER  XII. — The  synchronous  motor. 
CHAPTER  XIII. — The  rotary  converter. 
CHAPTER  XIV. — The  induction  motor. 
CHAPTER  XV. — Transmission  lines. 


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ADDITIONAL  WORKS  ON  PHYSICS 

FOR   REFERENCE  AND    CLASS  ROOM   USE 


ELECTRICITY  AND  MAGNETISM 

The  Principles  of  the  Transformer 

By  FREDERICK  BEDELL,  Ph.D.,  Cornell  University.         8vo.  Price  $3.25 

Magnetism  and  Electricity  for  Beginners 

By  H.  E.  HADLEY,  B.S.         Globe  8vo.  Price  60  cents 

A  Text-book  on  Elect ro= magnetism  and  the  Construc- 
tion of  Dynamos 

By  DUGALD  C.  JACKSON.         Vol.  I.  Price  $2.25 

'  Alternating  Currents  and  Alternating  Current  Machinery 

.?-    By  D.  C.  JACKSON  and  JOHN  PIERCE  JACKSON. 

Volume  II.  of  the  foregoing.          8vo.  Price  $3.50 

Electricity  and  Magnetism  for  Beginners 

By  F.  W.  SANDERSON,  M.A.         Globe  8vo.  Price  70  cents 

Practical  Physics.     Vol.  I. — Electricity  and  Magnetism 

By  BALFOUR  STEWART  and  W.  W.  GEE.         Globe  8vo.  Price  60  cents 

v   Elements  of  the  Mathematical  Theory  of  Electricity 
and  Magnetism 

By  J.  J.  THOMPSON,  F.R.S.         8vo.  Price  $2.60 

The  Storage  Battery 

By  AUGUSTUS  TREADWELL,  JR.         i2mo.  Price  $1.75 

>/ 

Theory  of  Electricity  and  Magnetism 

By  ARTHUR  GORDON  WEBSTER,  Clark  University.     8vo.  Price  $3.50 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 
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